iJTHE  NEW  INFINITE  AND 
THE  OLD  THEOLOGY 


By 
CASSIUS  J.  KEYSER,  PH.D.,  LL.D. 

Adrain  Professor  of  Mathematics  in 
Columbia  University 


NEW  HAVEN:  YALE  UNIVERSITY  PRESS 

LONDON  :  HUMPHREY  MILFORD 

OXFORD  UNIVERSITY  PRESS 


MDCCCCXV 

I 


COPYRIGHT,  1915 
BY  YALE  UNIVERSITY  PRESS 


First  printed  July,  1915,  1000  copies 


PREFACE 

Some  years  ago,  in  the  course  of  a  lecture 
dealing  with  Mathematics  regarded  as  a  dis- 
tinctive type  of  thought  and  with  its  rela- 
tions to  other  varieties  of  philosophic  and 
scientific  activity,  I  ventured  to  say:  "I  do 
not  believe  that  the  declined  estate  of  The- 
ology is  destined  to  be  permanent.  The 
present  is  but  an  interregnum  in  her  reign, 
and  her  fallen  days  will  have  an  end.  She 
has  been  deposed  mainly  because  she  has  not 
seen  fit  to  avail  herself  promptly  and  fully 
of  the  dispensations  of  advancing  knowledge. 
The  aims,  however,  of  the  ancient  mistress 
are  as  high  as  ever,  and  when  she  shall  have 
made  good  her  present  lack  of  modern  edu- 
cation and  learned  to  extend  a  generous  and 
eager  hospitality  to  modern  light,  she  will 
reascend  and  will  occupy  with  dignity,  as  of 
yore,  an  exalted  place  in  the  ascending 
scale  of  human  interests  and  the  esteem  of 
enlightened  men.  And  Mathematics,  by 
the  inmost  character  of  her  being,  is  espe- 
cially qualified,  I  believe,  to  assist  in  the 
restoration." 


iv  PREFACE 

The  following  pages  have  been  written 
under  the  stress  of  that  conviction,  which 
the  intervening  years  have  but  deepened  and 
confirmed.  Rational  theology  is  a  legitimate 
and  venerable  member  of  the  great  family  of 
spiritual  enterprises  of  man :  natural  science, 
philosophy,  jurisprudence,  religion,  art, 
mathematics,  theology.  These  are  all  of 
them  children  of  one  great  passion:  the 
imperious  craving  of  the  human  spirit  for 
an  inner  ideal  adjustment  of  life  to  the 
tragic  limitations  of  life  in  a  flowing  world. 
The  distinctive  problems  of  rational  theology 
are  regarded  as  in  a  special  sense  originating 
in  what  may  be  called  the  supernalizing 
tendence  or  power  of  the  human  mind.  This 
propensity  or  power,  so  strange  and  so  famil- 
iar in  every  category  of  the  understanding, 
ever  and  everywhere  manifesting  the  pres- 
ence of  a  kind  of  divine  energy  in  the 
world,  is  a  'natural'  agency,  being  at  once 
a  human  faculty  and  a  cosmic  force,  deeper 
than  will;  and  so  rational  theology  is  con- 
ceived to  be  a  species  of  'natural'  science — 
that  branch  of  it  which  has  for  its  special 
task  to  study  and  to  appraise  the  phenomena 
of  Idealization. 


PREFACE  v 

The  aim  has  been  to  set  the  matter  in  the 
increasing  light  of  certain  ideas  and  methods 
of  modern  mathematics.  But  the  reader 
need  not  be  deterred  by  any  fear  of  tech- 
nique. All  that  is  required  is  a  fair  share 
of  mathematical  spirit,  which  is  a  pretty 
common  possession,  being  simply  the  spirit 
of  right  thinking,  or  logical  righteousness. 

I  have  to  thank  the  Editor  of  the  Hibbert 
Journal  for  permission  to  employ  here,  in 
some  instances  with  only  slight  change,  a 
few  considerations  adduced  by  me  in  an 
article  published  in  that  journal  several 
years  ago  under  the  title,  "The  Message  of 
Modern  Mathematics  to  Theology." 

Columbia  University 
April  15,  1915 


THE   NEW   INFINITE   AND   THE 
OLD   THEOLOGY 


THE  NEW  INFINITE  AND  THE 
OLD  THEOLOGY 

It  is  the  aim  of  this  essay  to  show  that 
the  modern  concept  of  infinity  together  with 
certain  kindred  ideas  that  have  come  into 
mathematics  in  the  course  of  the  last  hun- 
dred years  have  qualified  this  science  to  shed 
new  light  upon  some  of  the  harder  problems 
of  rational  theology.  No  demand  will  be 
made  upon  the  reader's  knowledge  of  mathe- 
matical technique;  all  that  is  required  is  a 
fair  measure  of  mathematical  spirit,  which 
is  simply  the  spirit  of  logical  rectitude. 

The  reader  is  entitled  to  know  at  the  out- 
set that  the  following  words  are  not  those 
of  a  professional  theologian;  they  have  no 
official  authority,  nor  any  merit  beyond  what 
may  prove  to  be  their  reasonableness ;  they 
are  offered  as  the  words  of  a  layman  who, 
in  his  earlier  and  more  expectant  years, 
listened  attentively  to  some  hundreds  of 
sermons,  who  has  diligently  read  some  the- 
ological works,  and  has  reflected  a  good  deal, 
not  without  some  temperamental  interest  in 
the  themes,  upon  the  great  questions  that 


2  THE  NEW  INFINITE  AND 

attend  a  poignant  sense  of  the  world's  mys- 
tery and  wait  upon  the  leisure  hour  and  the 
pensive  mood. 

The  problems  are  many  and  difficult  and 
old.  No  one  who  has  seriously  reflected  upon 
them  or  is  familiar  with  their  history  will 
expect  to  find  in  these  pages  a  universal 
resolvent  for  theological  difficulties.  Prob- 
lems that  triumphed  over  the  keen  and 
sanguine  dialectic  of  the  ancient  world, 
problems  that  baffled  the  infinitely  subtle 
genius  of  the  middle  age,  problems  that  the 
profoundest  meditations  of  modern  philoso- 
phy have  not  been  able  to  solve,  present  grave 
difficulties.  Many  of  them  not  even  the 
adventurous  spirit  of  modern  mathematics 
may  confidently  assail.  My  task  is  limited 
to  showing  that  some  of  them  may  be  partly 
or  wholly  overcome  by  mathematical  means. 
That  all  the  rest  may  be  subdued  in  future 
by  similar  means  I  do  not  maintain.  Who 
knows?  It  may  be  that  some  of  the  difficul- 
ties are  insuperable  and  so  are  destined  to 
be  everlasting.  In  that  reflection  there  is 
nothing  to  bewail.  One  need  not  have 
"passed  on  life's  highway  the  stone  that 
marks  the  highest  point"  before  learning  to 


THE  OLD  THEOLOGY  8 

be  content  with  less  than  the  full  measure  of 
intellectual  conquest  dreamed  of  in  youth. 
To  be  happy  it  is  not  necessary  to  conquer 
the  invincible;  it  is  sufficient  to  advance  a 
little  where  progress  is  possible.  Indeed  it 
would  be  a  matter  for  sorrowing  if  in  the 
course  of  time  all  problems  were  solved  and 
questions  ceased  to  be,  for  a  world  without 
wonder  were  a  dreary  place.  But  of  that 
there  is  no  danger.  Wonder  increases  with 
knowledge  and  knowledge  with  time.  "It  is 
no  longer  true,"  said  Henri  Poincare,  "that 
there  are  solved  problems  and  others  that 
are  not  solved;  there  are  only  problems 
more  or  less  solved."  As  with  natural 
science  and  mathematics,  so  too  with  phi- 
losophy and  theology :  not  complete  solutions 
of  their  problems,  not  final  answers  to 
the  deepest  questionings  of  the  spirit,  but 
ever  increasing  illumination  of  them,  the 
acquisition  of  fresh  viewpoints  and  new 
perspectives — the  advancement,  in  a  word, 
and  multiplication  of  insight  and  vision — , 
these,  I  take  it,  are  the  reasonable  expecta- 
tions, the  precious  fruits,  the  ample  rewards 
of  serious  speculation. 

This,  then,  and  not  any  magical  formula 


4  THE  NEW  INFINITE  AND 

for  the  solution  of  riddles,  is  the  kind  of  ser- 
vice that  rational  theology  may  expect  from 
mathematics.  I  am  aware  that,  owing  to 
the  popular  misconception  of  mathematics, 
the  claim  is  not  an  easy  one  to  vindicate.  To 
the  many  who  are  accustomed  to  regarding 
mathematics  as  merely  a  useful  drudge,  the 
claim  will  naturally  seem  to  be  groundless 
or  visionary.  But  their  conception  of  the 
science  is  far  from  adequate  or  just.  Mathe- 
matics is  indeed  a  humble  servant — a  drudge, 
if  you  please — an  unsurpassed  drudge — in 
the  sense  that  nothing  else  does  a  larger 
share  of  humble  and  homely  work.  To 
imagine,  however,  that  her  place  in  the 
hierarchy  of  knowledges  is  thereby  defined 
is  hardly  the  beginning  of  wisdom  in  the 
matter.  It  is  necessary  to  look  much  higher. 
Her  rank  in  the  ascending  scale  is  not  that 
of  a  useful  drudge,  immeasurable  as  is  her 
service  in  that  capacity;  it  is  not  merely 
the  rank  of  a  metric  and  computatory  art, 
invaluable  as  the  latter  is  as  well  in  science 
as  in  the  affairs  of  the  workaday  world;  it 
is  not  even  that  of  servant  to  other  sciences 
in  their  fields  of  experimental  and  observa- 
tional research,  indispensable  as  mathematics 


THE  OLD  THEOLOGY  5 

is  in  that  regard;  over  and  above  these 
things,  she  is  charged  with  a  sacred  guard- 
ianship— in  her  keeping  are  certain  ideals, 
the  ideal  forms  of  science  and  the  standards 
of  perfect  thinking;  she  is  concerned,  not 
with  the  vagaries,  but  with  the  verities,  of 
thought,  with  select  matters  independent  of 
opinion,  passion,  accident,  and  will ;  it  is  thus 
peculiarly  hers  to  release  human  faculties 
from  the  dominion  of  sense  by  winning  alle- 
giance to  things  that  abide;  her  meditations 
transcend  the  accidents  of  time  and  place; 
it  is  their  idiosyncrasy  to  have  for  subject 
proper,  not  the  fickle  and  transitory  ele- 
ments in  the  stream  of  a  flowing  world,  but 
those  aspects  of  being  that  present  them- 
selves under  the  forms  of  the  infinite  and 
eternal. 

It  will  be  a  useful  preliminary  to  reflect 
a  little  upon  the  relations  of  rational  the- 
ology to  religion  on  the  one  hand  and  to 
science  on  the  other,  with  a  view  to  ascertain- 
ing what  the  province  of  theology  may  be 
rightly  said  to  be.  What,  then,  are  those 
relations?  It  is  evident  that  the  answer 
must  materially  depend  upon  the  relations 
that  science  and  religion  themselves  bear  to 


6  THE  NEW  INFINITE  AND 

one  another.  This  subject  I  have  discussed 
elsewhere.  In  a  recent  lecture*  dealing  with 
science  and  religion  I  have  undertaken  to 
examine  the  relations  between  these  two  great 
interests  of  mankind  from  what  may  perhaps 
be  regarded  as  in  some  respects  a  new  point 
of  view.  I  cannot  here  repeat  the  considera- 
tions adduced  in  support  of  the  doctrine 
there  sketched.  It  will  be  helpful,  however, 
and  possibly  sufficient,  to  set  down  briefly 
some  of  its  cardinal  propositions.  Those 
most  intimately  related  to  our  present  enter- 
prise are  these: 

A. — In  respect  of  method,  structure,  and 
content,  science  is  conceptual  and  logical. 
Any  branch  of  science,  at  any  given  stage 
of  its  development,  consists  of  a  certain 
group  of  ideas,  or  concepts,  together  with 
the  relations  that  bind  them  into  a  logically 
organic  whole.  The  potential  domain  of 
science  and  the  domain  of  the  rational — 
whatever  is  open,  that  is,  to  conquest  by  the 
means  of  concept  and  logic — are  one  and  the 
same.  All  else — whatever  is  below  or  above 
that  domain — is  subrational  or  superrational. 

*  Science  and  Religion:  the  Rational  and  the 
Superrational.  The  Yale  University  Press. 


THE  OLD  THEOLOGY  7 

B. — Religion,  on  the  other  hand,  is  not 
essentially  a  body  of  ideas  nor  a  body  of 
ideas  together  with  their  interrelations. 
Religion  is  essentially  and  ultimately  a  com- 
plex of  emotions,  of  emotions  as  felt  in  their 
integrity.  It  is  thus  a  kind  of  life  not  known 
nor  knowable  conceptually,  logically,  ration- 
ally, scientifically;  it  is  known  or  knowable 
only  "emotionally"  and  is  even  thus  know- 
able,  like  love,  for  example,  to  none  but  such 
as  feel  or  have  felt  the  constituent  emotions. 

C. — Religion  does  not  belong  to  the 
rational  domain.  There  is  indeed  possible 
a  science  of  emotions  but  these  can  not,  as 
emotions,  be  constituents  or  elements  of  it. 
For  it  they  do  not  exist  as  feelings.  It  can 
know  only  their  outward  manifestations  and 
can  know  these  only  as  science  may  know 
other  objects  of  the  external  world.  What 
is  called  the  scientific  study  of  religion  does 
not — as  scientific  it  can  not — deal  with  reli- 
gion as  "emotionally  known";  it  can  not 
know  religion  as  a  felt  life,  as  a  life  conscious 
of  itself;  the  most,  the  best,  the  last  it  can 
do  is  to  know,  as  objects,  as  externalities,  the 
exterior  manifestations  of  what  is  essentially, 
being  emotional,  an  inner  life.  Concepts  can 


8  THE  NEW  INFINITE  AND 

not  feel,  logic  can  not  fear  nor  love,  it  can 
not  revere,  wonder,  worship,  nor  adore.  For 
scientific  method  religion  is  not  a  life,  it  is 
an  hypothesis. 

D. — The  doctrine  that  it  is  a  character- 
istic mark  of  religion  "essentially  to  deal 
with  the  uncharted  region  of  human  expe- 
rience" is  untenable.  Ignorance  is  not  the 
presence  of  religion — else  every  body  would 
be  profoundly  religious — ,  it  is  the  absence 
of  knowledge.  Religion  and  the  spirit  of 
science  are  not  incompatible;  being  capable 
of  dwelling  together  harmoniously  in  a  single 
personality,  they  are  compatible  practically; 
and  they  are  compatible  theoretically:  under 
the  influence  of  advancing  science,  forms  of 
religion  age  and  pass  but  new  forms  succeed 
them,  and  the  religious  emotions  change  but 
they  do  not  die;  in  this  respect  it  is  with 
religion  as  with  knowledge — there  is  trans- 
formation and  supersession  of  form,  there  is 
advancement,  enlargement,  and  elevation, 
but  no  breach  of  continuity,  no  essential 
extinction,  no  death. 

E. — The  rational  implies  and  in  a  measure 
reveals  the  superrational.  The  rational 
world — the  potential  domain  of  science,  the 


THE  OLD  THEOLOGY  9 

field  of  concept  and  logic — is  not  the  whole 
sphere  of  our  psychic  life.  It  is  but  a  mid- 
region,  the  median  zone;  under  it  lies  a  sub- 
rational  zone — the  zone  of  sense,  which  we 
share  jointly  with  the  beasts;  above  it,  a 
world  superrational,  which  millions  have 
fancied  angels  share  with  us.  Though  it  is 
above  and  beyond  the  dominion  of  concept 
and  logic,  the  existence  of  that  world  is  yet 
betrayed  and  its  nature  in  part  displayed, 
by  rational  means:  by  a  process  known  in 
mathematics  as  the  method  of  limits  but 
elsewhere  known  as  the  process  of  idealiza- 
tion. Operating  amid  the  activities  of  con- 
cept and  logic  and  upon  their  subject- 
matter,  the  great  process  occurs  in  every 
division  of  the  rational  understanding;  its 
function  is,  in  every  category  where  the  laws 
of  reason  reign,  to  point  aloft  to  an  appro- 
priate limit  beyond  their  range,  to  some  ideal 
form  above  the  laws :  in  the  category  of 
classes,  to  an  ideal  universe  as  the  manifold 
of  all;  in  the  realm  of  propositions  or  that 
of  relations,  to  the  sum  or  the  product  of 
all  propositions  or  all  relations ;  in  that  of 
time,  to  eternity;  in  knowledge,  to  omnis- 
cience; in  ubiety,  to  omnipresence;  in  power, 


10          THE  NEW  INFINITE  AND 

to  omnipotence;  in  order  and  law,  to 
necessity  or  fate;  in  indetermination,  to 
absolute  freedom  or  self-determination;  in 
wisdom  or  love,  to  the  "beauty  absolute"  of 
Plato's  dream ;  and  so  on  and  on  throughout 
the  circuit  and  scope  of  rational  thought. 
And  so  it  is  that  the  realm  of  superrational 
reality — the  ultimate  source  of  the  religious 
emotions — thus  indicated  by  the  supernal- 
izing  process  of  idealization  operating  in  the 
fields  of  reason,  presents  itself  as  an  over- 
world  of  ideals. 

In  view  of  these  considerations  respecting 
the  relations  of  Science  and  Religion,  what 
shall  we  say  is  the  place  of  Theology  ?  What 
are  the  essential  relations  of  Theology  to 
Science  and  to  Religion?  What  and  where 
is  the  province  of  the  venerable  "Queen"? 
It  is  evident,  I  think,  that  theology  has  a 
province.  Certainly  there  is  in  the  heart  of 
mankind  a  perennial  craving  for  a  kind  of 
wisdom  that  the  ages  have  taught  us  to 
regard  as  the  peculiar  object  of  theological 
aspiration;  and  there  is  a  corresponding 
realm  of  truth:  a  field  of  enquiry  that  the- 
ology may  rightly  claim  as  her  own.  We  are 
in  a  position,  I  believe,  to  see  pretty  clearly 


THE  OLD  THEOLOGY  11 

what  her  province  is  and  what  it  is  not.  I 
speak,  of  course,  of  rational  theology.  No 
one  need  be  told  nowadays  that  theology  is 
not  religion.  Religion,  we  have  seen,  is  essen- 
tially and  ultimately  a  certain  complex  of 
emotions — of  emotions,  not  as  analyzed  into 
their  elements,  but  as  felt  in  their  native 
integrity.  Religion,  so  taken,  is  not  only 
more  immediate  and  more  fundamental  than 
theology  but  differs  from  it  in  kind :  theology 
is  not  emotion,  it  is  doctrine.  No  doubt 
religion  is,  in  a  sense,  pregnant  with  the- 
ology, containing  it,  so  to  speak,  in  a  "state 
of  solution",  in  potentia;  theology  thus  is, 
in  a  sense,  religion's  offspring  and  is  natur- 
ally pervaded  and  tinged  by  religious  refer- 
ence and  feeling.  But  to  confound  or  to 
identify  the  two  things  would  be  like  confus- 
ing a  doctrine  of  aesthetic  with  the  sentiment 
of  beauty,  or  an  ethical  theory  with  the 
sense  of  right  and  wrong,  or  mathematical 
science  with  a  feeling  for  logical  implication 
and  intellectual  harmony,  or  science  in 
general  with  the  feeling  of  wonder,  the 
delight  of  understanding,  the  lure  of  truth, 
the  joy  of  knowledge  and  light.  All  doc- 
trine, all  theory,  all  science  results  from  the 


12          THE  NEW  INFINITE  AND 

reaction  of  intellect  to  feeling.  "Gefiihl" 
may  not  be  "alles"  but  it  is  back  of  all  and 
under  all.  The  emotional  source,  however, 
or  background  of  a  doctrine  is  not  itself 
doctrine.  A  dogmatist  may  feel,  but  dogmas 
are  not  emotions,  they  are  propositions. 
Rational  theology,  in  order  to  be  rational, 
must  be  an  affair  of  intellect,  it  must  be  an 
affair  of  ideas  and  their  relations,  of  con- 
cepts and  logic ;  it  must  be  scientific — scien- 
tific in  subject-matter,  in  method,  and  in 
structure;  and  so  must  deliver  its  message 
in  the  form,  not  of  poetry  or  song  or  ejacu- 
lation, but  of  reasoned  propositions  concate- 
nated into  an  intelligible  and  coherent  body 
of  doctrine  addressed  primarily  to  the 
understanding. 

If  theology  is  to  be  thus  regarded  as  a 
science,  what  shall  we  say  is  its  subject- 
matter?  What  is  theology  the  science  of? 
The  answer  is  hardly  to  be  found  in  the 
etymological  meaning  of  the  term.  Names 
are  stabler  than  their  meanings.  Time  is 
ever  pouring  new  wine  into  old  bottles  but 
the  bottles  do  not  always  burst.  Geometry, 
as  every  one  knows,  is,  etymologically,  earth- 
measurement,  and  the  corresponding  term 


THE  OLD  THEOLOGY  18 

in  the  Chinese  language  means  'show  it  by  a 
figure.'  Geometricians  know,  however,  that 
their  science  is  not  mainly  concerned  with 
measurement  of  any  kind,  much  less  with 
measurements  of  Earth,  and  they  know  that, 
far  from  depending  on  figures,  which  are 
things  of  sense  and  imagination,  geometry 
is  a  purely  conceptual  architecture,  always 
strictly  and  for  the  most  part  obviously 
transcending  sense  and  imagination.  The- 
ology may  indeed,  in  the  future  as  in  the 
past,  discourse  about  gods  or  about  God; 
she  may  do  so  legitimately,  conveniently, 
often  consistently  with  an  immense  literature 
and  a  vast  tradition.  But  to  speak  of  "a 
science  of  God",  even  if  the  locution  were 
clear,  which  it  is  not,  could  hardly  serve  as 
a  felicitous  indication  of  the  subject-matter, 
or  the  field,  of  a  science.  It  does  not  ring 
right:  it  sounds  extravagant,  pretentious, 
irreverent.  In  so  far  as  God  must  be  sup- 
posed to  be  superrational,  the  speech  is 
absurd;  and  it  has,  moreover,  the  fatal 
disadvantage  of  seeming  to  exclude  from  the 
circle  of  theological  thought  the  finest  spir- 
itual meditations  of  many  millions  of  our 
fellow  men.  If  we  insisted  upon  defining  the 


14          THE  NEW  INFINITE  AND 

province  of  theology  etymologically,  a 
devout  adherent  of  so  great  and  noble  a 
religion  as  Buddhism,  for  example,  however 
profound  his  understanding  of  spiritual 
things,  would  have  to  be  denied,  and  he 
would  disclaim,  the  interests  and  the  charac- 
ter of  theologian.  Such  a  conception  is 
shallow  and  narrow.  The  domain  of  what  is 
to  be  called  theology  must  be  conceived  with 
sufficient  depth  and  catholicity  to  include  the 
thought  of  all  men  and  women,  whatever  their 
time,  place  or  creed,  whose  vocation  it  is  to 
cherish  the  kind  of  wisdom  that  seeks  to 
understand  and  to  interpret  rationally  the 
supreme  ideals  of  the  human  spirit. 

Shall  we,  then,  say  that  theology  is  the 
science  of  religion?  There  are,  I  think, 
insuperable  objections  to  doing  so.  For  one 
thing,  the  words  have  been  gradually  appro- 
priated in  recent  times  to  another  use;  they 
carry  a  different  import ;  they  point  to  some- 
thing else.  They  point,  on  the  one  hand,  to 
psychological  analysis  of  the  religious  emo- 
tions and,  on  the  other,  to  study  of  their 
external  manifestations — to  their  sensible 
embodiments  in  institutions,  customs,  cere- 
monies, and  rites.  Such  analysis  and  such 


THE  OLD  THEOLOGY  15 

study  are  important  enterprises;  they  are 
intimately  related  to  theology ;  in  a  measure, 
they  fall  within  its  scope,  but  only  so  as 
auxiliaries  and  adjuvants  and  not  as  con- 
stituting the  center  or  bulk  of  its  concern. 
Not  only  do  they  differ  from  theology  in 
their  attitude  towards  religion,  being  less 
warm,  less  sympathetic,  less  constructive, 
less  philosophic  in  their  interest  and  bearing, 
less  interior  and  spiritual,  but — what  is  more 
significant — they  differ  from  it  in  respect  of 
content  and  subject-matter.  Theology  may 
analyze  the  religious  emotions,  or  try  to  do 
so,  and  it  may  study  their  exterior  mani- 
festations in  time  and  place,  but  these  enter- 
prises are  not,  singly  or  jointly,  its  chief 
concern.  Theology  is  neither  a  branch  of 
analytic  psychology  nor  a  branch  of  anthro- 
pology nor  yet  a  combination  of  them.  What 
is  called  the  science  of  religion — the  anthro- 
pological study  of  religion — is  related  to 
religious  life  very  much  as  botany  would  be 
related  to  the  life  of  plants  if  we  supposed 
plants  to  be  conscious  of  what  we  call  their 
life  and  if  botanists  were  fairly  repre- 
sentative vegetables.  But  before  botany 
could  develop  a  branch,  or  acquire  an  inter- 


16          THE  NEW  INFINITE  AND 

est  or  a  function,  analogous  to  that  of 
theology,  it  would  be  necessary  to  endow 
plant  life  much  more  highly  than  we  have 
just  supposed.  We  should  have  to  suppose 
it  endowed  with  fear  and  love,  reverence  and 
awe,  hope  and  aspiration,  with  supernalizing 
power,  with  dreams  and  ideals,  with  respon- 
sive sensibility  to  the  light  of  a  higher 
world.  The  relation  of  theology  to  religion 
is,  then,  not  that  of  a  science  to  its  subject- 
matter.  Granted  that  religion  is  theology's 
source,  its  motive,  reference,  and  goal,  its 
raison  d'etre.  That  does  not  mean  that  reli- 
gion is  its  subject-matter.  We  have  seen 
that  religion  is  essentially  emotion;  theology 
is  doctrine;  the  former  feels;  the  latter 
thinks ;  theology  is  a  structure — an  edifice 
of  thought;  religion  is  a  flow — a  stream  of 
sentiment;  theology  is  subject  to  the  govern- 
ance of  ideas,  deriving  its  authority  from 
the  rules  of  reason;  religion  is  under  the 
sway  of  ideals,  deriving  its  authority  from 
reason's  dreams;  the  materials  of  the  former 
are  near  at  hand,  they  belong  to  the  domain 
of  the  rational;  the  emotions  of  the  latter 
come  from  afar,  having  their  ultimate  source 
in  a  realm  super  rational ;  the  light  of  the- 


THE  OLD  THEOLOGY  17 

ology  is  the  light  of  the  understanding ;  that 
of  religion  is  the  mystic  radiance  of  an  over- 
world. 

In  this  triune  scheme  of  distinct  but 
kindred  things  of  the  spirit,  in  this  triple 
combination  and  interplay  of  idea,  ideal,  and 
feeling — of  reason,  overworld,  and  reli- 
gion— ,  it  is  now  at  length  evident,  I  believe, 
where  we  are  to  find  the  province  and  the 
role  of  theology.  It  is  evident  that  its  part 
in  the  great  drama  is  the  part  of  idea  and 
reason,  the  part  of  intellect.  The  subject- 
matter  of  theology  is  not  immediately  nor 
primarily  the  religious  emotions  nor  is  it  the 
interior  constitution  of  their  superrational 
ground  and  source:  it  is  neither  the  feelings 
themselves  nor  the  essential  inner  nature  of 
the  overworld  that  thrills  them  into  being 
and  sustains  their  life.  Its  subject-matter 
proper  consists  of  rational  phenomena:  it 
consists  of  those  facts  and  processes  of  the 
rational  understanding  that  serve  at  once  to 
indicate  the  existence  of  an  overworld  and 
to  manifest  its  shining  upon  the  things  below. 
It  is  thus  the  task  of  theology  to  study  those 
implications  of  logical  thought  that  are 
hyperlogical,  and,  in  so  far  as  possible,  to 


18          THE  NEW  INFINITE  AND 

interpret  them  in  rational  terms;  it  is  its 
function  to  examine  the  nature  of  rational 
thinking  in  its  various  categories,  to  unfold 
its  hid  intent,  to  clarify  the  manner  in  which 
thought,  following  endless  courses  within  its 
own  domain,  perpetually  approximates,  for- 
ever pursues,  and  intimates,  by  the  laws  of 
its  going,  limits  that  lie  beyond.  There  is 
in  Reason  a  life-process  deeper  and  finer  than 
the  mechanical  movements  of  ratiocination, 
there  is  a  kind  of  divine  energy  there,  a 
beholding  presence,  a  faculty  within  a  fac- 
ulty, a  soul,  if  you  please,  in  Reason,  that 
fills  her  heart  with  dreams,  points  to  a  shin- 
ing canopy  above  the  summits  of  her  thought, 
discerns  in  the  light  and  atmosphere  of  her 
common  activities  the  sheen  of  ideals — the 
glory  of  perfections — above  and  beyond 
them  all.  The  nature  and  significance  of 
that  supernalizing  power — there,  I  take  it, 
is  theology's  problem.  Theology  is,  in  a 
word,  the  science  of  Idealization. 

It  is  a  natural  science ;  not  indeed  a  labor- 
atory science;  not,  in  ordinary  sense,  an 
observational  science,  for  the  objects  it  ob- 
serves are  inner  things,  things  beheld  only 
in  psychic  light,  not  things  stained  with 


THE  OLD  THEOLOGY  19 

refracted  radiance  of  the  sun;  neither  is  it 
an  experimental  science  save  in  the  sense  in 
which  all  thinking  whatsoever — all  logical 
procedure — is  essentially  experimental;  but 
it  is,  none  the  less,  a  natural  science. 
Granted  that  its  materials  are  not  things  of 
sense;  granted  that  they  are  things  of 
reason — familiar  shinings  there  of  strange 
supernal  lights ;  they  are  not  on  that  account 
unnatural.  The  phenomena  of  idealization 
are  not  artificial  nor  forced;  they  are  spon- 
taneous, springing  from  foundations  deeper 
than  will;  they  are  as  natural  as  the  dawn; 
their  credentials  are  cosmic. 

What  it  is  that  makes  the  task  of  theology 
so  difficult  and  delicate  is  clearly  to  be  found 
in  the  peculiar  character  of  its  subject- 
matter — in  the  essential  nature  of  it  and 
especially  in  the  relation  it  bears  to  the  over- 
world.  We  have  to  do  with  the  phenomena 
of  idealization.  There  are  no  special  diffi- 
culties to  be  encountered  in  dealing  with  the 
great  process  as  a  process;  the  major  fact 
about  it — that  of  its  existence,  its  ubiquitous 
presence,  its  ceaseless  operation — is  plain; 
there  is  nothing  insuperable  in  ascertaining 
where  and  how  it  begins — here,  there  and 


20          THE  NEW  INFINITE  AND 

yonder,  as  we  have  seen — in  every  category 
of  the  rational  understanding;  nor  in  ascer- 
taining how  it  advances,  from  initial  points 
in  the  domain  of  reason  along  innumerable 
paths  of  thought  that  run  endlessly  on  and 
on,  like  an  increasing  sequence  of  terms, 
outward  towards  the  border;  nor  yet  in 
ascertaining  how  the  process  ends,  in  the 
presentation,  namely,  of  limits  that  lie 
beyond.  In  all  this,  in  all  that  pertains  to 
the  process  as  such,  there  are  indeed  difficul- 
ties, subtleties  of  thought,  delicate  considera- 
tions, but  nothing  of  a  kind  to  baffle  the 
methods  of  science.  But  what  shall  we  say 
of  the  results  of  the  process,  of  the  limits 
presented  by  it,  of  those  great  ideals  them- 
selves of  which  it  is  the  function  of  ideali- 
zation to  make  us  aware?  It  must  be  noted 
and  borne  in  mind  that  they  are  not  con- 
cepts, they  are  not  ideas,  they  are  ideals. 
How  is  theology,  how  is  theology  as  a 
science,  to  deal  with  them?  Their  similitudes 
and  differences  are  to  be  detected;  they  are 
to  be  compared,  ordered,  and  classified; 
their  significance  is  to  be  appraised;  their 
authority  determined;  their  claim  to  su- 
premacy in  the  ascending  scale  of  values 


THE  OLD  THEOLOGY  21 

must  be  examined.  How  may  all  this  be 
done  scientifically?  In  such  an  enterprise 
the  primal  instinct  to  seize  and  subjugate  is 
of  no  avail.  Ideals  are  not  things  to  be 
grasped,  they  are  things  to  be  reached  for; 
they  are  not  subjects  for  conquest,  they 
are  objects  for  aspiration;  they  are  not 
properties  to  be  possessed,  they  are  perfec- 
tions to  be  pursued;  logic  can  not  harness 
them,  it  can  not  reduce  them,  as  it  reduces 
ideas,  to  the  ranks  of  obedient  servants  in 
the  fields  of  reason;  they  hover  aloft;  they 
can  not  be  pounced  upon ;  to  realize  an  ideal 
is  not  to  possess  it ;  it  is  to  own  its  authority, 
to  respond  to  its  appeal,  to  follow  its  leading, 
to  be  drawn  to  higher  elevations  by  the 
charm  and  persuasiveness  of  its  majesty  and 
beauty.  It  is  evident,  I  believe,  what  must 
be  the  answer  to  the  foregoing  question..  In 
dealing  with  the  great  ideals,  theology  must 
approach  them  from  below,  from  their 
ground  and  source,  which  are  a  rational 
ground  and  source ;  she  must  approach  them 
through  an  understanding  of  the  infinite 
sequences  that  have  the  ideals,  not  as  final 
terms,  in  reason,  but  as  superrational  limits ; 
she  can  know  them  only  as  they  are  revealed 


22          THE  NEW  INFINITE  AND 

in  the  mode  and  light  of  their  genesis ;  she 
must  study  them  as  results  of  a  process — 
results  that  she  can  not  immediately  handle 
or  seize — a  process  with  which  she  is  com- 
petent so  to  deal. 

An  even  greater  source  of  theological 
difficulty  and  confusion  is  the  subtle  and 
bewildering  relation  the  ideals  in  question 
bear  to  the  overworld.  Theology  is  to  be 
rational,  scientific.  The  overworld  is  super- 
rational.  It  is  obvious  that  such  a  world 
can  not  be  the  subject  of  a  science.  It  is 
evident  that  theology  can  not  be  a  science 
of  the  overworld  as  astronomy,  for  example, 
is  the  science  of  the  heavenly  bodies,  as 
physics  is  the  science  of  matter  and  motion, 
as  biology  is  the  science  of  organic  life,  or 
as  mathematics  is  the  science  of  logical 
implication.  To  speak  of  explaining  super- 
rational  being  in  rational  terms  is  folly. 
Does  it,  therefore,  follow  that  theology 
must  remain  silent  regarding  superrational 
reality?  It  does  not.  The  overworld  has 
downward-facing  aspects  ;  it  presents  aspects 
to  the  upward  gaze  of  reason:  these  are 
reason's  ideals,  superrational  limits,  as  we 
have  seen,  of  rational  thought.  Of  these 


THE  OLD  THEOLOGY  28 

theology  may  speak;  she  may  speak  of  their 
origin,  of  the  process  and  mode  of  their 
presentation,  of  their  significance,  of  their 
majesty,  of  the  lure  of  their  beauty,  of  their 
glory;  she  may  speak  of  their  genuineness 
and  authority,  of  their  relation  to  hope  and 
aspiration,  yearning  and  love,  reverence  and 
awe.  But  she  can  not  without  folly  under- 
take to  explore  nor  pretend  to  explain  the 
inner  constitution,  the  ultimate  nature,  of 
an  overworld. 

The  task  of  theology,  thus  conceived,  is 
one  of  exceeding  delicacy.  It  is  little  wonder 
that  in  her  long,  long  career  she  has  often 
gone  astray,  that  she  has  committed  innu- 
merable blunders,  that  she  has  sometimes 
despaired,  that  she  has  frequently  incurred, 
sometimes  deservedly,  the  disrespect,  the 
antipathy,  even  the  contempt,  of  scientific 
men.  It  is  little  wonder,  too,  that  she  fares 
ill  in  a  practician  age,  that  she  wins  but 
little  encouragement  or  support  in  compari- 
son with  those  physical  sciences  that  have 
the  advantage  of  being  able  constantly  to 
vindicate  their  worth  in  the  eyes  of  a  tinker- 
ing and  huxtering  world  through  "useful" 
applications,  multiplying  the  conveniences  of 


24          THE  NEW  INFINITE  AND 

men,  advancing  their  physical  welfare,  ex- 
panding and  subliming  their  petty  pursuits 
to  the  proportions  and  elevation  of  vast  and 
dazzling  commercial  and  industrial  enter- 
prises. There  is  no  domain  of  thought,  no 
branch  of  science  or  speculation,  where  the 
subject-matter  is  quite  so  subtle,  where  the 
facts  are  so  intangible,  so  elusive,  so  remote 
from  sound  and  touch  and  sight,  where  the 
conceptions  are  so  tenuous,  where  the 
hypotheses  are  so  generic  and  broad,  so 
hard  to  verify,  and  where  it  is  so  difficult 
to  discriminate  appearance  from  reality, 
separating  from  out  the  wildering  maze 
problems  that  are  genuine  from  those  that 
are  not.  It  is  precisely  on  this  account, 
however,  that  modern  mathematics,  as  I  hope 
we  may  see,  is  qualified  by  the  inmost  char- 
acter of  her  being  to  lend  a  helping  hand. 

The  answer  of  Laplace  to  Napoleon's 
question,  why  he  had  not  in  his  Mecanique 
Celeste  mentioned  the  name  of  God,  is  known 
to  all:  "Sir,"  the  savant  replied,  "I  had  no 
need  of  that  hypothesis."  Not  so  generally 
known  is  the  instant  response  of  the  great 
author  of  the  Mecanique  Analytique  when 
the  Emperor  made  prompt  report  to  him 


THE  OLD  THEOLOGY  25 

of  the  memorable  conversation:  "Neverthe- 
less," said  Lagrange,  "that  is  an  hypothesis 
that  accounts  for  many  things." 

Let  us  not  mistake  the  point  of  these  fine 
words.  Superficially  the  speeches  appear  to 
be  mutually  antagonistic ;  they  do  somewhat 
resemble  the  sudden  saber-thrust  and  counter 
thrust  of  battle.  Yet  they  are  in  perfect 
accord.  Their  semblance  of  mutual  oppo- 
sition is  illusion,  due  to  the  dramatic  char- 
acter of  the  situation  and  a  certain  contrast 
of  sound.  It  entirely  disappears  on  closer 
examination.  There  is  neither  irreverence  in 
the  one  speech  nor  reverence  in  the  other.  If 
Laplace's  mot  indicate  a  lack  of  veneration, 
then  that  of  Lagrange  must  indicate  a  lack 
of  scientific  temper.  Scientific  temper  lack- 
ing in  Lagrange !  It  is  true  that  Laplace, 
at  the  close  of  his  immortal  work,  might,  like 
Newton  before  him,  have  discharged  the 
mood  essential  to  its  production;  he  might 
have  given  himself  to  another  kind  of  medi- 
tation, to  leisured  contemplation  of  the 
cosmic  visions  gained  in  years  of  analytic 
toil;  and  thus  receptively  musing  on  the 
mighty  mechanism  of  the  stellar  universe — 
its  unfathomable  deeps,  the  immeasurable 


26          THE  NEW  INFINITE  AND 

energies  of  swift- revolving  worlds  of  flame, 
the  all-pervasive  order,  the  silent  reign 
throughout  of  majestic  law — ,  he  might 
have  felt  a  reverent  sense  of  admiration  akin 
to  religious  awe,  and — again  like  Newton — 
have  owned  in  words  that  such  unity  and 
power  betoken  the  dominion  of  a  Supreme 
Ruler  and  Lord  of  all.  Had  he  done  so,  had 
he  thus  chosen  to  crown  his  scientific  work 
by  some  expression  of  belief  in  a  divine 
source  and  ruler  of  a  universe  whose  pro- 
founder  beauties  he  had  been  enabled  to 
behold  and  disclose,  the  testimony  could  not 
but  seem  fitting  to  everyone;  it  would  be 
especially  grateful  to  those  fortunate  folk 
who  see  in  every  great  display  of  power  a 
witness  to  omnipotence,  in  every  striking 
manifestation  of  natural  law  an  evidence  of 
divine  decree,  in  every  nobler  scene  of  beauty 
a  token  of  divine  perfection.  But — and  this 
is  the  thing  to  be  noted — such  an  expression 
of  belief,  however  creditable  to  the  great 
astronomer  in  his  character  as  a  man,  would 
not  have  been  in  any  sense  a  constituent  of 
the  Mecanique  Celeste — neither  a  postulate 
nor  a  theorem,  no  proper  part  whatever  of 
the  great  description,  but  only  an  after- 


THE  OLD  THEOLOGY  27 

effect,  a  note  of  veneration  evoked  by  subse- 
quent recall  and  contemplation  of  the  celes- 
tial scenes  described.  Had  some  soldier  of 
Euclid's  time  demanded  of  the  illustrious 
geometrician  why  he  had  not  in  the  Elements 
made  mention  of  Zeus,  no  doubt  the  wit 
provoked  but  yesterday  by  the  challenge  of 
Napoleon's  question  had  framed  itself  in 
Greek  two  thousand  years  before.  Or  does 
some  one  imagine  that  that  least  perishable 
work  among  the  scientific  monuments  of  the 
ancient  world  could  have  been  scientifically 
improved  by  adding  to  its  underlying  postu- 
lates the  statement,  There  is  a  God?  If  one 
asks,  for  example,  why  planetary  paths  are 
elliptic,  or  why  camels  have  humps,  or  why 
the  earth  is  flattened  at  the  poles,  and 
receives  for  answer  that  there  is  a  God  and 
that  God  so  wills,  the  answer  may  indeed  be 
a  statement  of  fact,  and  yet  as  a  scientific 
answer  it  would  be  absolutely  worthless;  it 
would  be  silly ;  and  any  one  who  could 
solemnly  offer  it  as  scientific  would  seem  less 
logical  than  pathological.  The  resolute 
attempt  of  science  to  explain  the  universe 
in  terms  of  mechanics  can  not  be  furthered 
by  the  postulation  of  a  God ;  indeed  it  would 


28          THE  NEW  INFINITE  AND 

be  abandoned  thereby;  for  one  thing  is 
certain :  God,  if  God  there  be,  is  no  machine. 
Laplace  was  right;  he  had  "no  need  of  that 
hypothesis."  Nay,  his  problem  being  one  of 
mechanics,  he  could  not,  without  stultifying 
himself,  have  even  pretended  to  use  it. 

"Sir,  I  had  no  need  of  that  hypothesis." 
Laplace  was  right.  "Nevertheless  that  is  an 
hypothesis  that  accounts  for  many  things." 
Lagrange  was  right.  It  is  evident  that  the 
significance  of  the  two  speeches  lies,  not  in 
their  seeming  discord,  but  in  their  real  con- 
cord: in  their  common  point  of  view;  it 
consists  in  what  neither  one  asserts  but  both 
of  them  imply:  namely,  that  God  is  an 
hypothesis. 

Let  me  say,  for  what  it  may  be  worth, 
that  personally  I  am  far  from  prepared  to 
contend  that  God  is  the  name  of  an  hypothe- 
sis and  nothing  more.  It  is  perfectly  true 
and  perfectly  clear  that  science,  viewed  as 
an  attempt  to  explain,  in  mechanical  terms, 
all  phenomena,  the  attempt  itself  included, 
is,  thoroughgoingly,  an  atheistic  enterprise. 
It  is  a  legitimate  enterprise;  it  is  carried  on 
under  a  working  hypothesis  that  men  may 
make — the  hypothesis  that  mechanical  prin- 


THE  OLD  THEOLOGY  29 

ciples  are  sufficient;  under  it  great  things 
have  been  achieved;  there  is  every  reason  to 
expect  that  even  greater  things  will  follow 
with  the  years;  it  is  a  right,  it  may  be  a 
duty,  to  pursue  it  for  all  it  may  yield.  But, 
while  Science,  thus  defined,  is  essentially 
atheistic,  scientific  Man  is  not.  Man  is 
greater,  infinitely  greater,  than  science,  as 
he  is  greater  than  art  or  philosophy  or 
religion  or  any  mode  or  form  in  which  his 
life  may  manifest  itself.  Many  a  scientific 
man  is  temperamentally  disqualified  to  re- 
gard the  mechanistic  hypothesis  as  all- 
sufficient,  and  who  is  qualified  to  say  that 
temperament  has  no  essential  relation  to  the 
problem?  Many  a  scientific  man,  even  the 
hardiest  of  the  kind — unless  cut  off  before 
the  mellowing  touch  of  pensive  years  can 
ripen  knowledge  into  wisdom — comes  sooner 
or  later  to  feel  that  the  mechanistic  hypothe- 
sis, fruitful  as  it  is,  can  not  embrace  the 
whole  of  life,  that  it  can  never  give  an  ade- 
quate account  of  the  finer  elements  of  "man's 
unconquerable  mind" — its  radiance  and  joy, 
its  conscience  and  love,  its  spiritual  yearn- 
ings, its  holy  aspirations;  and  so,  under  the 
chastening  influences  of  time  and  meditation, 


30          THE  NEW  INFINITE  AND 

more  and  more  awake  to  the  subtler  claims 
of  his  being,  he  comes,  reluctantly  perhaps, 
slowly  it  may  be  and  late  in  life,  to  reconsider 
and  rectify  his  earlier  estimates,  and  from 
the  doubt  that  is  "hungry  and  barren  and 
sharp  as  the  sea,"  craves  and  seeks  relief, 
finding  it  at  length  in  a  sense  of  a  sympa- 
thizing consciousness  not  his  own,  in  subtle 
intimations  of  the  pervasive  presence  of  a 
living  Spirit. 

Neither  do  I  deny  that,  far  from  being  a 
mere  hypothesis,  God  may  be  a  real  being — 
an  infinite  personality — whose  reality  is,  at 
times,  to  persons  of  a  certain  temperament, 
an  immediate  object  of  a  genuine  kind  of 
knowledge — the  kind  that  mystics  have 
sometimes  claimed  to  have.  That  they  have 
been  sincere  there  is  no  reason  to  doubt. 
Have  they  been  mistaken?  I  do  not  know. 
Knowledge  of  the  kind  in  question  is  said 
to  be  ineffable.  If  it  exists,  it  is  ineffable. 
That  does  not  mean  that  it  does  not  exist; 
it  merely  means  that,  if  it  does  exist,  it  is  not 
scientific  knowledge,  for  scientific  knowledge 
is  effable:  it  is  communicable  knowledge;  it 
rests  on  a  kind  of  evidence  that,  if  it  is  for 
you,  is  also  for  me,  or,  if  for  me,  then  also 


THE  OLD  THEOLOGY  81 

for  you — it  is  not  essentially  private  or 
personal,  it  is  essentially  public  and  imper- 
sonal. But  knowledge,  we  know,  is  not  all 
of  a  kind.  It  would  be  stupid  to  maintain 
that  all  knowledge  must  be  scientific  or  else 
ungenuine.  I  know  how  to  move  my  arms, 
we  say,  or  how  to  walk,  to  cast  a  stone,  to 
wink,  to  swallow,  or  to  think;  a  squirrel,  we 
all  say,  knows  how  to  climb  a  tree  or  gnaw 
a  nut,  a  horse  how  to  find  its  stall:  such 
knowledge  is  not  scientific.  To  my  wife  the 
full  moon  appears  the  size  of  a  dinner  plate; 
to  me,  the  size  of  a  large  cart-wheel.  How 
big  is  the  moon?  That  is  not  the  question; 
if  it  were,  the  right  answer  would  belong  to 
scientific  knowledge.  How  big  does  it  seem? 
That  is  meaningless.  How  big  does  it  seem 
to  you?  That  you  can  know.  To  me? 
That  I  know.  But  your  certitude  and  mine 
are  not  common  to  us — they  are  not  imper- 
sonal, they  are  individual,  private,  personal 
certitudes. 

In  this  connection  it  is  worth  while  to 
mention  another  type  of  evidence — a  kind 
of  evidence  that  is,  like  the  mystic's,  in- 
effable— at  all  events  exceedingly  hard  to 
communicate — and  yet  is,  I  suspect,  avail- 


32          THE  NEW  INFINITE  AND 

able  to  the  normal  intellect,  provided  it  will 
be  at  the  pains  to  try  a  certain  psychological 
experiment.  The  experiment  relates  to  the 
great  question  of  cosmic  purposefulness. 
To  deny  the  universe  that  quality  is  so  easy 
to  do  in  words.  But  to  do  so  in  fact — to 
gain,  that  is,  a  poignant  sense  of  the  denial's 
essential  meaning — appears  to  be  a  matter 
of  exceeding  difficulty.  May  I  refer  to  my 
own  experience?  I  have  tried  the  experiment 
many  times.  Mood  is  essential,  and  time  and 
place — springtime  or  autumn,  evening  or  the 
still  night,  rural  solitude  under  the  moon 
and  the  stars.  In  the  course  of  thirty  years 
I  have  won,  perhaps  a  hundred  times,  what 
seemed  to  be  a  realizing  sense  of  what  it  is 
that  the  denial  means.  Words  fail.  To 
know  the  sense  one  must  feel  it.  When  it 
comes,  it  comes  like  a  sudden  apparition,  but 
it  does  not  stay.  Its  momentary  presence 
seems  to  involve  an  instant's  failure  of  its 
support,  like  a  swooning  of  mind  immediately 
checked  and  healed,  like  the  integrity  of 
being  itself  suddenly  recovered  from  the 
brink  of  dissolution.  The  coming  and  going 
are  quick  as  a  streak  of  lightning;  only  the 
apparition  is  dark,  like  the  passing  shadow 


THE  OLD  THEOLOGY  38 

of  a  flitting  bird,  like  a  mid-day  moment's 
dream  of  dusk  at  once  dissolved  in  the  light, 
like  a  cut  in  consciousness  instantly  closed 
as  a  cleft  in  a  sea :  the  denial  being  no  sooner 
achieved  in  feeling  than  it  has  been  com- 
pletely overwhelmed  by  the  inrushing  flood 
of  'What,  then,  is  it  for?' — as  if  some  sud- 
denly roused  instinct,  vital  to  Intelligence, 
had  leaped  to  the  defense  of  her  integrity 
and  life.  Such  experience  leads  me  to  sus- 
pect that  cosmic  purposefulness  is  something 
profounder  than  a  doctrinal  postulate  with 
which  thought  may,  if  it  choose,  dispense. 
I  suspect  it  is  an  essential  part  of  what  mind 
means  by  mind. 

But,  after  all  such  claims  have  been  duly 
allowed,  we  must  not  fail  to  see  clearly  that, 
for  theology  regarded  as  a  purely  scientific 
enterprise,  God  is  an  hypothesis  and  nothing 
more.  For  the  rapt  vision  of  the  seer,  faith's 
evidence  of  things  not  seen,  the  mystic's 
immediate  sense  of  divine  communion,  the 
above-mentioned  evidence  of  cosmic  purpose- 
fulness,  all  these  and  their  kind  being  essen- 
tially personal,  private,  ineffable,  incommu- 
nicable experiences,  are  none  of  them  forms 
of  scientific  knowledge:  because,  as  I  have 


84          THE  NEW  INFINITE  AND 

said,  scientific  knowledge  always  is,  poten- 
tially at  least,  impersonal,  public,  sharply 
discriminated  in  kind  from  other  varieties  of 
knowledge  by  what  we  may  call  its  social 
character,  by  its  transmissibility  from  mind 
to  mind.  Knowledge  of  the  outward  differ- 
ences between  Greek  architecture,  for  ex- 
ample, and  Hindu  architecture  is  scientific 
but  your  knowledge  that  one  of  these  pleases 
you  more  than  the  other  is  not  scientific,  it 
is  private.  The  idiosyncrasy  of  scientific 
knowledge  is  that,  though  perchance  there 
may  be  at  a  given  time  but  one  individual 
who  has  it,  yet  it  does  not  belong  to  him  in 
his  individual  capacity ;  it  is  his  as  a  member 
of  society,  as  a  representative  of  humankind. 
Of  expert  logicians,  for  example,  there  may 
be  in  a  given  community  but  few.  Yet  the 
science  is  not  theirs.  Logic  is  impersonal. 
It  belongs  to  Man. 

Here,  then,  we  are  face  to  face  with  a 
capital  theme  of  theological  meditation:  the 
assumption,  namely,  or  hypothesis  of  a  being 
called  God.  How  shall  we  frame  it  in  speech? 
How  describe  the  august  Being  it  seeks  to 
represent?  If  we  appeal  to  the  greatest 
physical  philosopher  of  all  time,  the  author 


THE  OLD  THEOLOGY  85 

of  the  Principia  and  inventor  of  the  Infinit- 
esimal Calculus  returns  the  terse  reply: 
"A  Being  eternal,  infinite,  absolutely  per- 
fect." If  we  listen  to  him  whose  genius 
established  the  great  alliance  between  the 
doctrines  of  Number  and  Space,  thus  bring- 
ing together  the  sundered  hemispheres  of 
apodictic  thought  and  so  creating  the  world 
of  Analytic  Geometry,  we  hear  the  resound- 
ing words  of  Descartes:  "Infinite,  eternal, 
immutable,  independent,  all-knowing,  all- 
powerful."  If  we  ask  the  "God-intoxicated" 
philosopher  of  Amsterdam,  we  receive  from 
the  great  Spinoza  a  similar  characterization 
not  less  impressive:  "Absolutely  infinite, 
consisting  of  infinite  attributes,  each  express- 
ing eternal  and  infinite  essentiality."  These 
familiar  citations  will  serve  to  remind  the 
reader  of  like  efforts,  among  the  best  of 
human  thought,  to  formulate  in  adequate 
terms  the  hypothesis  God.  About  things 
that  are  very  familiar  it  is  exceedingly 
difficult  to  bring  ourselves  to  think,  and  the 
terms  of  the  hypothesis  have  been  familiar 
for  hundreds  of  years.  Were  it  new  and 
fresh  instead  of  being  so  old  and  stale,  we 
should  all  of  us  be  immediately  struck  by 


36          THE  NEW  INFINITE  AND 

what  is,  among  its  distinctive  features,  a 
very  obvious  mark.  The  hypotheses  that  we 
meet  elsewhere,  as  the  nebular,  the  corpus- 
cular, the  ionic,  the  atomic,  the  molecular, 
the  hypothesis  of  a  space-pervading  ether, 
of  universal  gravitation,  of  Euclidean 
space,  of  organic  evolution,  of  conservation 
of  energy  or  of  mass,  all  such — all  the 
hypotheses  we  encounter  in  the  literature  of 
ordinary  science — have  one  character  in 
common:  each  of  them  is  restricted  in  scope, 
limited  to  some  fragment  of  reality,  they 
divide  in  order  to  conquer,  each  is  confined 
to  a  field  that  is  bounded;  not  so,  however, 
the  hypothesis  God:  it  is  distinguished  by 
the  fact  that,  among  hypotheses,  it  alone 
attempts  to  span  and  bind  the  Whole.  That 
is  a  very  remarkable  characteristic.  And 
soon  we  must  note  another.  But  first  we 
must  ask,  What  does  the  hypothesis  mean? 

"The  light  of  human  minds,"  says  Hobbs, 
"is  perspicuous  words,  but  by  definitions 
first  snuffed  and  purged  from  ambiguity." 
Accordingly  it  is  necessary  to  ask:  what,  if 
any,  precise  meaning,  available  for  the  pur- 
poses of  logical  discourse,  may  be  assigned 
to  the  terms  of  the  hypothesis?  Infinite, 


THE  OLD  THEOLOGY  87 

Eternal,  Omnipotent,  Omniscient,  Omni- 
present, and  the  rest:  what  do  these  mighty 
terms  mean?  I  do  not  now  ask  for  their 
meaning  as  instruments  for  energizing  life. 
I  do  not  now  ask  for  their  meaning  as  cries 
of  the  spirit — voices  from  the  deeps  of  feel- 
ing. At  present  I  am  not  concerned  with 
their  meaning  for  reverence,  for  love,  for 
awe.  I  do  not  here  seek  their  relation  to  the 
moods  of  poetry  and  prayer.  I  am  not  en- 
quiring for  their  emotional  significance,  so 
like  that  of  mountain  scenery,  a  vast  wilder- 
ness, the  heavens  above,  or  the  "solemn 
anthem  of  the  sea."  I  enquire  for  their 
logical  value,  for  their  meaning  in  Thought. 
It  is  essential  to  note  at  once  a  very  remark- 
able thing:  the  great  terms  in  question  are 
not  names  of  scientific  notions,  they  are  not 
names  of  concepts,  they  are  not  names  of 
ideas;  they  are  names  of  ideals — super- 
rational  ideals,  outlying  limits  of  rational 
thought.  This  gives  the  hypothesis  an  ap- 
pearance of  being  an  hypothesis  respecting 
the  nature  of  the  overworld.  Is  it  such  in 
fact?  So  to  take  it,  as  many  consciously  or 
unconsciously  do,  is  fatal.  It  removes  the 
question  from  the  jurisdiction  of  theology 


38          THE  NEW  INFINITE  AND 

regarded  as  a  science:  a  world  of  super- 
rational  reality  can  not  be  the  subject  of 
any  science;  to  suppose  the  contrary  is  to 
ignore  the  sole  restriction  that  the  muses 
have  placed  on  freedom  of  thought:  thought 
must  be  free  from  internal  contradiction,  it 
must  be  harmonious.  It  is  not  necessary, 
however,  to  take  the  hypothesis  so.  The 
overworld,  we  have  seen,  has  downward- 
facing  aspects.  These  aspects,  presented  to 
the  upward  gaze  of  reason  by  the  process 
of  idealization  operating  in  reason's  fields, 
are  precisely  the  ideals  that  the  terms  of  our 
hypothesis  serve  to  designate.  The  hypothe- 
sis is  accordingly  to  be  regarded  as  an 
hypothesis  respecting,  not  the  essential  inner 
nature  of  the  overworld,  but  the  nature  of 
the  downward-facing  aspects  presented  by  it 
through  the  process  of  idealization — a  super- 
nalizing  agency  working  below.  But,  if  the 
ideals,  the  aspects  in  question,  are  super- 
rational,  how  is  it  possible  for  science  to  say 
aught  about  them?  Science,  we  have  seen, 
must  approach  them  from  below,  in  the  light 
of  their  genesis  and  manner  of  presentation. 
By  this  method  science  is  enabled  to  say  of 
them  that  such  and  such  ideals  are  of  a 


THE  OLD  THEOLOGY  39 

nature  to  appear  as  limits  of  such  and  such 
processes  or  sequences  of  rational  thought. 
To  be  able  to  say  that,  however,  with  all  it 
implies,  is  much:  just  as  it  is  much — if  I 
may  illustrate*  great  things  by  small — to  be 
able  to  say  of  a  curve,  for  example,  that, 
though  outside  the  domain  of  broken  lines, 
it  is  yet  the  limit  of  an  endless  sequence  of 
broken  lines;  or  just  as  it  is  much  to  be  able 
to  contemplate  a  curved  surface  as  an  ideal 
indicated  by  an  endless  series  of  plane- 
bounded  figures  approximating  it  forever, 
though  the  ideal  itself  does  not  belong  to  the 
field  of  the  approximating  figures;  or  just 
as  it  is  much  to  be  able  to  view  what  is  called 
an  irrational  number  as  an  ideal  or  limit 
beyond  the  domain  of  rational  numbers  but 
indicated  and  endlessly  pursued  by  series  of 
these;  or  just  as,  in  general,  it  is  important 
for  the  life  of  understanding,  to  be  able  to 
make  out,  in  whatever  field  it  operates,  the 
endless  courses  of  ever  increasing  approxi- 
mation that  by  the  law  of  their  progress  at 
once  betray  appropriate  perfections  beyond 

*  For  a  fuller  explanation  of  the  force  and  point 
of  such  illustrations,  see  the  author's  "Science  and 
Religion,"  herein  cited  on  an  earlier  page. 


40          THE  NEW  INFINITE  AND 

and,  though  never  attaining  them,  yet  lead 
us  more  and  more  deeply  into  their  far- 
shining  light. 

Such,  then,  must  be  the  method  and  such 
the  ways  of  rational  theology.  If  it  is  to 
have  a  motto,  the  motto  must  be:  From 
ideas  to  ideals.  The  former  indicate,  the 
latter  are  indicated.  These  are  to  be  under- 
stood scientifically  only  in  so  far  as  their 
meaning  is  revealed  in  the  ideas  indicating 
them  and  in  the  manner  of  the  indication. 
Among  the  ideals  with  which  we  are  here 
concerned,  among  the  great  ideals  assembled 
in  the  hypothesis  God,  it  is  obvious  that  there 
is  one  which  has  the  distinction  of  seeming 
to  be  at  once  coordinate  with  the  rest  and 
yet  in  a  sense  involved  in  each  of  them.  I 
refer,  of  course,  to  the  ideal  denoted  by 
the  adjective  Infinite.  There  is  a  Being, 
so  the  hypothesis  runs,  at  once  infinite  and 
omniscient  and  omnipotent  and,  so  on. 
Omniscience,  however,  involves  Infinitude; 
so  does  Omnipotence;  so  does  Eternality; 
so  does  every  pealing  note  of  the  great 
diapason.  Any  illumination  of  what  is 
meant  by  the  term  Infinite  will,  therefore, 


THE  OLD  THEOLOGY  41 

serve  in  a  measure  to  illuminate  the  meaning 
of  the  kindred  terms. 

The  reader  doubtless  knows  that  the 
Infinite  of  theology  has  never  been  defined — 
defined,  that  is,  for  logical  use — ,  and  he  is 
now  in  a  position,  I  believe,  to  see  why  it  has 
not.  It  is  not  because  the  centuries  have  not 
witnessed  many  ingenious  attempts  to  define 
it.  It  is  because  the  term  denotes,  not  an 
idea,  but  an  ideal,  a  superrational  ideal,  and 
so  does  not  admit  of  definition.  There  can 
be  no  doubt  that  a  great  deal  of  the  confusion 
found  in  theological  literature  has  resulted 
from  the  fact  that  theologians,  failing  in 
respect  of  this  logical  distinction,  have  gone 
on  discoursing  about  what  they  have  called 
the  Infinite,  as  if  the  term  stood  for  some- 
thing— a  concept  or  an  idea — that  had  been, 
or,  at  all  events  could  be,  defined.  The 
remedy  for  the  kind  of  confusion  that  thus 
results  is  simple:  it  consists  in  not  ignoring 
the  distinction;  it  consists  in  ceasing,  once 
for  all,  the  attempt  to  treat  an  ideal  as  an 
idea ;  it  consists  in  refraining  from  the  hope- 
less endeavor  to  deal  with  a  superrational 
limit  as  if  it  were  a  rational  term  of  the 
endless  sequence  whose  nature  it  is,  not  to 


42          THE  NEW  INFINITE  AND 

contain  the  limit   nor   to   attain   it,   but   to 
indicate  it  and  approximate  it. 

There  are,  however,  other  difficulties  con- 
nected with  the  term — difficulties  inherent 
in  the  nature  of  the  case,  proper  difficulties, 
we  may  say,  because  they  belong  to  the  ideas 
constituting  what  we  may  call  the  ideal's 
rational  basis,  its  basis  in  reason.  Here  it 
is  necessary  to  note  an  important  distinction, 
to  be  henceforth  kept  in  mind.  Thus  far  we 
have  been  speaking  of  the  theological  infinite. 
Fortunately  or  unfortunately  the  same  term 
is  constantly  employed  in  science  and  espe- 
cially in  mathematics  in  another  sense — in 
a  sense  closely  related  indeed,  as  we  shall 
see,  to  theology's  sense  of  the  term  but  yet 
quite  distinct  therefrom.  In  theology,  we 
have  seen,  the  term  denotes  an  ideal,  a  super- 
rational  ideal,  which  can  not  be  defined;  in 
mathematics,  as  we  are  going  to  see,  it  de- 
notes an  idea,  a  concept  that  not  only  is 
sharply  definable  but  has  been  in  fact  sharply 
defined.  Presently  we  shall  begin  to  see  the 
beautiful  relation  between  the  two  senses  in 
which  the  term  is  employed  and  why  it  is 
and  wherein  the  mathematical  sense  is  an 
indispensable  means  for  making  clear  the 


THE  OLD  THEOLOGY  48 

theological  sense.  The  ideas — the  objects 
or  things — that  mathematics  calls  infinite 
are  not  all  of  them  of  one  order.  There  are 
countless  mathematical  infinities,  or  infini- 
tudes, or  infinites,  as  they  are  variously 
called, — countless  types  of  them.  Like  the 
stars,  they  differ  in  glory.  They  constitute, 
as  we  shall  see,  an  endless  sequence  of  ever 
increasing  terms — an  endless  series  or  suc- 
cession of  terms  mounting  ever  higher  and 
higher  in  respect  of  order  or  dignity  or  rank. 
Each  term  in  the  endless  march  of  terms 
includes  the  type  of  infinitude  represented 
by  the  preceding  term  but  is  itself  of  higher 
type.  The  major  relation  in  the  scheme 
delineated  is  evident  at  once :  the  limit  of  this 
endless  series  of  infinite  ideas  is  an  infinite 
ideal:  the  Infinite  of  theology  is  the  limit  of 
the  endless  sequence  of  more  and  more 
embracing  Infinitudes  presented  by  science. 
It  is  not  one  of  them ;  it  is,  so  to  speak,  their 
envelope,  enfolding  them  all. 

It  is  now  time  to  look  into  the  great  rela- 
tion a  little  more  deeply.  We  must  try  to 
see  quite  clearly  what  scientific  infinities  are ; 
we  must  endeavor  to  understand  how  they 
are  related  to  the  finite  things  of  sense  and 


44          THE  NEW  INFINITE  AND 

how  they  embrace  and  penetrate  the  common 
affairs  of  men;  we  must  learn  something  of 
the  law  in  accordance  with  which  they  are 
disposed,  rank  above  rank,  in  a  hierarchy  of 
orders,  without  a  summit;  we  must  observe 
how  the  process  of  Idealization,  operating 
in  and  among  them,  pervades  the  atmosphere 
of  the  grand  array,  and  creates  or  finds  there 
a  subtle  radiance  that  seems  to  reveal  a  down- 
ward-shining aspect  of  an  overworld.  The 
task  is  not  an  easy  one ;  it  demands  a  little 
patience  and  a  little  penetration;  a  part  of 
the  discussion,  which  is  for  such  as  prefer 
not  being  entertained  to  being  fooled,  must 
seem  to  some  a  little  arid — a  pretty  dry  way 
to  a  valley  of  fruits;  a  sense  of  its  full 
significance  can  not  be  gained  at  once;  it 
must  be  won  as  the  fruit  of  reflection. 

In  order  to  explain  the  scientific  or 
mathematical  meaning  of  the  term  infinitude, 
or  infinite,  let  me  begin  with  some  simple 
examples.  I  will  take  them  from  the  two 
hemispheres  of  rigorous  thought,  the  two 
great  subject-matters  of  it — Number  and 
Space. 

Imagine  two  concentric  spheres,  the  inner 
one  white  and  named  the  silver  sphere,  the 


THE  OLD  THEOLOGY  45 

outer  (or  larger)  one  yellow  and  named  the 
golden  sphere.  (In  accordance  with  the 
usage  of  higher  geometry  I  shall  mean  by 
sphere  a  sphere-surface.)  Next  imagine  the 
sheaf  (as  it  is  called)  of  rays,  consisting  of 
all  the  straight  lines  that  have  their  begin- 
ning at  the  common  center  of  the  two  spheres 
and  thence  extend  outward  endlessly  in  every 
direction.  It  is  plain  that  any  ray,  R,  of  the 
sheaf  pierces  the  silver  sphere  in  a  point, 
say  S,  and  the  golden  sphere  in  a  point,  say 
G.  Calling  S  and  G  a  pair  of  points,  it  is 
evident  that,  by  considering  all  the  rays  of 
the  sheaf,  the  points  of  the  one  sphere  are 
paired  with  those  of  the  other  in  a  one-to- 
one,  or  point-to-point,  fashion:  in  other 
words,  a  unique  and  reciprocal  correspond- 
ence is  thus  established  between  the  points 
of  the  silver  sphere  and  those  of  the  golden 
sphere.  One  silver  point  corresponds  to  one 
and  but  one  golden  point;  one  golden  point, 
to  one  and  but  one  silver  point;  and  this 
reciprocal  relation  holds  for  every  silver 
point  and  for  every  golden  one.  We  see  at 
once  that  the  number  of  points  on  the  one 
sphere  is  exactly  the  same  as  the  number 
of  points  on  the  other;  we  see,  too,  that  this 


46          THE  NEW  INFINITE  AND 

number  equality  subsists  no  matter  how  great 
the  difference  between  the  sizes  of  the 
spheres — one  of  them  may  as  well  be  micro- 
spically  small  and  the  other  billions  of  times 
larger  than  the  earth  or  the  sun.  In  other 
words,  the  number  of  points  on  a  surface  of 
given  size  (given  area)  is  independent  of  the 
given  area  or  size,  and  so  will  not  be  changed 
by  changing  the  area  or  size.  Now  imagine 
a  closed  curve  or  ring — red,  if  you  like,  for 
the  sake  of  vividness — to  be  drawn  on  the 
golden  sphere  and  enclosing  thereon  a  por- 
tion of  it,  a  region  A,  precisely  equal  in  area 
to  the  area  of  the  silver  sphere.  We  need 
not  suppose  this  latter  equality  (of  areas) 
but  we  may  as  well,  for  the  supposition  will 
reduce  a  little  the  shock  we  are  soon  to 
receive.  The  number  of  points  in  the  region 
A  is,  of  course,  the  same  as  the  number  on 
the  silver  sphere  and  is,  therefore,  the  same 
as  the  number  on  the  golden  one.  But  the 
collection  of  points  in  the  region  A  is  only 
a  part  of  the  whole  collection  on  the  golden 
sphere.  The  shocking  thing  is  this :  we  have 
here  a  part — the  ensemble  of  points  in  the 
region  A — and  a  whole — the  ensemble  of 
points  on  the  golden  sphere — such  that  the 


THE  OLD  THEOLOGY  47 

number  of  points  in  the  part  is  precisely  the 
same  as  the  number  of  points  in  the  whole. 
It  is  to  be  noted  carefully  and  once  for  all 
that  the  astonishing  equality  subsists,  not 
between  the  area  of  the  region  A  and  that 
of  the  golden  sphere,  but  between  two  multi- 
tudes (of  points),  of  which  one  is  a  part  and 
the  other  the  whole. 

Does  some  non-mathematical  reader,  un- 
familiar with  this  kind  of  thinking,  distrust 
the  argument,  feeling  perhaps  that  he  has 
been  tricked  by  a  juggling  use  of  the  notions 
of  surface  area  and  point  collection?  If  so, 
let  him  scrutinize  the  equivalent  following 
argument,  in  which  the  notion  of  area  plays 
no  role.  First,  a  preliminary  word  of  expla- 
nation. If  a  straight  line  and  a  plane  are 
parallel,  we  say  sometimes — in  high  school, 
for  example, — that  they  have  no  common 
point ;  but  if  we  continue  our  study  into  what 
is  called  projective  geometry, — never  mind 
the  name — ,  we  learn  to  say  that,  if  a  line 
and  a  plane  be  parallel,  they  have  a  common 
point — called  an  "ideal"  point  to  distinguish 
it  from  ordinary  points — the  "ideal"  point 
being  so  far  away  that  it  can  not  be  reached 
by  a  step-by-step  process  of  going  towards 


48          THE  NEW  INFINITE  AND 

it;  any  "ordinary"  point  can  be  so  reached. 
Such  "ideal"  points  of  a  plane  make  up  a 
line,  called  the  "ideal"  line  of  the  plane. 
A  plane  thus  conceived  as  having  such  an 
"ideal"  line  is  called  a  projective  plane,  and 
a  line  regarded  as  having  an  "ideal"  point 
is  called  a  projective  line.  And  now  the 
promised  argument.  Think  of  a  hemisphere, 
H,  and  suppose  it  to  rest  on  a  horizontal 
plane,  77,  the  hollow  of  H  being  open  to  the 
upper  sky.  The  rim  of  H  is  a  circle,  C. 
Denote  its  center  by  P.  There  is  a  sheaf 
of  rays  running  out  from  P.  Of  this  sheaf 
consider  only  those  rays  that  lie  in  the  plane 
containing  C  and  those  that  run  below  this 
plane.  The  imagery  is  perfectly  clear.  The 
rays  considered,  since  each  of  them  pierces 
H  in  a  point  and  H  in  a  point,  plainly  estab- 
lish a  one-to-one  correspondence  between  the 
points  on  the  hemisphere  H  and  the  points 
of  the  plane  77;  a  point  on  the  rim  of  H 
obviously  corresponding  to  an  "ideal"  point 
of  77,  and  conversely.  Now  imagine  a  plane 
above  77,  parallel  to  it,  and  cutting  H  in  two. 
Cast  away  the  upper  part  of  H  so  cut  off 
and  keep  the  lower  part — the  up-turned  cap 
resting  on  77  as  before.  The  rim  of  this  cap 


THE  OLD  THEOLOGY  49 

is,  again,  a  circle.  Call  it  C'  and  its  center 
P'.  As  before,  there  is  a  sheaf  of  rays  run- 
ning out  from  P'.  Of  this  sheaf  consider, 
as  before,  only  those  rays  that  lie  in  the 
plane  of  the  cap's  rim  and  those  that  run 
below  this  plane.  Again  the  situation  is 
clear:  the  rays  considered,  each  piercing  a 
point  from  the  cap  and  a  point  from  77,  set 
up  a  one-to-one  correspondence  between  the 
points  of  the  cap  and  those  of  77.  We  now 
see  that  the  number  of  points  of  7:7  is  the 
same  as  the  number  of  points  of  77  and  that 
the  number  on  the  cap  is  the  same  as  the 
number  on  77;  hence,  we  see,  the  number  on 
H  is  the  same  as  the  number  on  the  cap. 
But  the  collection  of  points  on  the  cap  is  a 
part  of  the  whole  collection  on  77.  The  fact 
is,  accordingly,  now  perfectly  evident — 
whether  at  first  we  like  it  or  not — that  we 
are  in  a  world  where  it  is  easy  to  encounter 
a  whole  having  a  part  whose  elements  are 
precisely  as  numerous  as  are  the  elements 
of  the  whole.  Every  whole  of  that  kind  is 
said  to  be  infinite. 

Be  it  understood,  then,  that  the  concept 
of  infinity — the  scientific  or  mathematical 
meaning  of  the  term — is  this:  namely,  a 


50          THE  NEW  INFINITE  AND 

collection,  class,  set,  group,  aggregate, 
ensemble,  manifold,  or  multitude  of  elements 
or  things — be  these  points  or  passions,  ions 
or  ideas,  relations  or  terms,  quantities  or 
qualities,  numbers  or  instants  or  colors  or 
sounds,  degrees  of  wisdom  or  goodness  or 
power  or  joy,  or  any  other  modes,  forms, 
or  determinations  of  being — is  said  to  be 
infinite  if  and  only  if  the  collection,  like  the 
ensemble  of  points  on  a  sphere,  contains  a 
part,  or  subcollection,  that  is  numerically 
equal  to  the  whole.  On  the  other  hand,  a 
collection  or  multitude  is  said  to  be  finite 
if  and  only  if,  like  the  collection  of  trees  in 
yonder  forest,  like  the  human  population  of 
the  globe,  like  the  multitude  of  sands  of  the 
sea  or  that  of  the  stars  within  telescopic 
range,  it  contains  no  part  or  subcollection 
numerically  equal  to  the  whole. 

There  is  here  no  ground  for  quibbling, 
hesitance,  or  doubt.  There  stand  the  two 
concepts,  absolutely  clear;  and  there,  too, 
stand  the  validating  facts,  absolutely  unmis- 
takable. The  latter  indeed  may  be  multi- 
plied at  will.  Examples  of  collections 
illustrating  the  concept  of  finitude  are  of 
course  familiar  to  every  one,  being  forced 


THE  OLD  THEOLOGY  51 

upon  the  attention  by  the  vulgar  necessities 
of  life;  indeed  they  are  so  familiar  that  but 
few  persons  have  so  much  as  dreamed  that 
there  are  collections  or  manifolds  of  another 
type.  We  have  seen,  however,  that  there  are, 
and  the  gain  is  one — if  we  really  make  it  our 
own — to  work  a  profound  transformation  in 
our  view  of  the  world.  Of  examples  illus- 
trating the  concept  of  infinitude,  we  have 
thus  far  instanced  but  two.  Similar  examples 
abound,  however,  in  even  greater  profusion 
than  the  other  kind,  being  found  in  the  great 
and  the  small,  the  remote  and  the  near,  in 
Number,  in  Space,  in  Time,  in  qualitative 
distinctions,  in  the  realm  of  pure  relation — 
wherever  the  intellect  may  penetrate — if  the 
inner  eye  be  only  disciplined  to  detect  their 
omnipresence.  A  little  patience,  I  have  said, 
is  indispensable  in  this  part  of  the  discus- 
sion— quite  as  needful  as  a  little  penetration ; 
and  I  must  request  the  reader's  permission 
to  tarry  yet  a  little  in  order  to  point  out  a 
few  further  illustrations  of  what  science 
means  by  an  infinite  multitude.  It  is  of  the 
nature  of  doctrine  to  grow  aloft,  higher  and 
higher,  into  the  limpid  atmosphere  of  pure 
theory,  and  that  is  legitimate — architecture 


62          THE  NEW  INFINITE  AND 

must  rise;  but,  however  high  its  head,  a  doc- 
trine, if  it  is  to  stand,  must  plant  its  feet 
upon  the  solid  earth  of  fact.  The  facts  with 
which  we  are  here  concerned  are  not  facts  of 
sense;  they  are  facts  of  thought;  they  do 
not  belong  to  the  domain  that  we,  as  animals, 
share  jointly  with  the  beasts ;  they  are  the 
prerogatives  of  man  as  man — in  his  capacity, 
that  is,  for  "discourse  of  reason."  Let  us 
return  for  a  moment  to  our  image  of  the 
sheaf  and  the  concentric  spheres.  Consider 
those  rays  of  the  sheaf  that  pierce  the  points 
of  the  region  A  on  the  golden  sphere.  Let 
us  call  the  bunch  of  these  rays  a  bundle. 
It  is  evident  that  the  number  of  rays  of  the 
bundle  is  the  same  as  the  number  of  points 
in  the  region  A,  one  ray  through  each 
point  of  A,  one  point  of  A  on  each  ray  of  the 
bundle:  this  number,  we  have  seen,  is  the 
same  as  the  number  of  points  on  the  sphere ; 
and  this,  again,  the  same  as  the  number  of 
rays  of  the  entire  sheaf;  whence  it  is  seen 
that  the  bundle,  though  but  a  part  of  the 
sheaf,  has  the  same  number  of  rays  as  the 
number  of  rays  in  the  whole.  And  so  the 
sheaf  and  the  bundle  serve  to  exemplify 
again  the  concept  of  infinite  manifolds. 


THE  OLD  THEOLOGY  58 

Let  me  now  take  a  very  simple  example 
from  the  inexhaustible  resources  of  another 
field.  Consider  the  little  equation,  y  =  2x, 
which  every  one  understands.  If  we  assign 
a  value,  say  1,  to  #,  then,  as  we  see,  the  value 
of  y  is  thereby  also  determined:  it  is  just 
twice  as  much — in  this  case  2 ;  if  we  let 
x  be  ^,  y  must  be  1 ;  if  x  be  V2,  y  is  2  V2 ; 
and  so  on:  to  any  value  of  the  variable  x, 
there  corresponds  one  and  but  one  value  of 
the  variable  y;  and  conversely,  for  we  could 
just  as  well  give  values  to  y  and  so  find  for 
each  of  them  its  half,  or  the  corresponding 
value  of  x.  Let  us  now  agree  to  let  x  vary 
in  value  from  zero  to  1,  taking,  one  at  a  time, 
the  value  zero,  the  value  1,  and  each  of  the 
innumerable  host  of  values  between;  then  y 
will  take,  one  at  a  time,  each  of  the  values 
in  the  range  from  zero  to  2,  including,  of 
course,  zero  and  2.  Thus  is  set  up  a  one-to- 
one  correspondence  between  the  multitude 
of  values  in  the  #-range  from  zero  to  1  and 
the  multitude  in  the  ^/-range  from  zero  to  2. 
The  number  of  values  or  numbers  in  the  one 
range  is,  therefore,  the  same  as  the  number 
of  values  or  numbers  in  the  other.  But  the 
collection  of  numbers  in  the  range  from  zero 


54          THE  NEW  INFINITE  AND 

to  1  is  but  a  part  of  the  whole  multitude  in 
the  range  from  zero  to  2.  Accordingly  each 
of  these  multitudes  is  an  infinite  multitude 
of  things. 

As  a  final  example  here,  let  me  invite  care- 
ful attention  to  an  infinite  collection  that 
would  be  the  easiest  of  all  to  grasp  were  it 
not  so  very  simple  and  if  action  upon  it 
of  the  higher  understanding  were  not  almost 
inhibited  by  our  fixed  habit  of  regarding  the 
elements  of  the  collection  as  having  no  sig- 
nificance beyond  their  familiar  vulgar  uses 
in  the  counting-house  and  the  market  place. 
For  the  elements  in  question  are  nothing 
more  romantic  than  the  numbers  with  which 
we  count.  How  very  prosaic  the  prospect, 
you  naturally  say.  I  quite  agree,  and  yet  I 
venture  to  say  that,  if  we  will  but  rise  above 
the  stale  levels  of  sense  and  imagination,  we 
shall  not  fail  to  detect  here  a  species  of  genu- 
ine poesy — the  poesy  of  pure  thought  in 
touch  with  the  infinite  and  eternal.  Consider 
the  two  sequences  or  series  of  integers : 

(W)     1,2,3,4,5,6,       ....,»,  n  +  1, 

(P)      2, 4,  6,  8, 10, 12,  .  .  .  .  ,  <2n,  2(n  +  1) 

By  the  series  (  W)  of  symbols  I  wish  to  call 
attention,  not  to  that  uncompleted  row  of 


THE  OLD  THEOLOGY  55 

marks  itself,  but  to  a  certain  definite  invisible 
whole  that  the  row  suggests  and  serves  to 
bring  as  an  object  before  the  mind,  namely: 
the  totality  of  the  positive  integers.  On 
being  confronted  with  the  notion  of  this 
fundamental  totality,  at  once  so  clear  to 
thought  and  so  baffling  to  imagination,  many 
persons,  especially  the  uninitiated,  become 
restive  for  a  time.  A  little  reflection,  how- 
ever, will  dissipate  any  reasonable  scepticism, 
and  show  that  our  footing  here  is  solid  rock. 
It  is  true  indeed  that,  however  many  integers 
we  may  singly  specify  or  imagine,  there 
always  remain  more  and  more.  It  is  also 
true  that  the  hand  cannot  actually  write  nor 
the  physical  eye  behold  a  set  of  symbols 
matching  one-to-one  all  the  integers  com- 
posing the  asserted  totality,  if  such  a  thing 
there  be.  What  of  it?  Consider,  for  a 
moment,  a  familiar  totality  so  obvious  that 
none  may  question  it — the  totality,  I  mean, 
of  the  points  of  a  circle.  As  in  the  case  of 
the  integers,  so  here,  too,  it  is  impossible  to 
think  all  the  points  singly  or  singly  to  specify 
or  symbolize  them  all.  Yet  there  they  are — 
not  one  now  and  then  another — but  all  of 
them  at  once,  a  totality  persisting  as  such 


56          THE  NEW  INFINITE  AND 

and  unescapable.  What  is  the  secret?  The 
secret  is  that  the  totality  is  a  conceptual 
thing,  a  thing  for  thought  and  not  for  sense 
or  imagination,  a  thing  carved  out  by  a  law 
transcending  the  powers  of  step-by-step 
perception  or  depiction,  a  law  of  definition 
that  selects  out  of  the  universe  of  thinkable 
things  a  set  of  them  unambiguously — the 
law,  namely,  that  the  things  shall  be  points 
of  a  plane  and  be  all  of  them  equally  distant 
from  a  point  therein.  So  it  is  precisely  with 
the  totality  of  positive  integers.  If  you  say 
that  the  totality  does  not  exist,  what  you 
mean  is  that  the  integers  of  such  a  totality 
can  not  be  written  down  for  sight  to  look 
at  or  that  no  one  can  depict  them  all  on  the 
canvas  of  imagination.  Permit  me  to  remind 
you  that  I  am  not  here  addressing  your 
sense  nor  your  imagination.  I  am  addressing 
your  conception,  your  thought.  The  asserted 
totality  does  not  exist  for  sense,  it  does  not 
exist  for  imagination;  it  exists  for  thought. 
It  derives  its  completeness  and  one-ness  from 
the  completeness  and  one-ness  of  the  selective 
law  defining  it — the  law,  namely,  that  after 
any  definite  integer  there  is  another  greater 
than  that  by  one.  Note  that  the  law  includes 


THE  OLD  THEOLOGY  57 

and  excludes  and  that  the  inclusion  and 
exclusion  are  both  of  them  precise,  decisive, 
complete,  and  instantaneous.  It  is  pathetic 
if  one  can  not  see  clearly  that  it  is  precisely 
such  sense-transcending  and  imagination- 
transcending  totalities  that  constitute  the 
essential  subject-matter  of  rigorous  thought. 
For  to  deny  their  validity  is  to  evacuate  the 
Reason  of  its  proper  content  and  to  bar  even 
the  possibility  of  Science.  Science,  properly 
speaking,  does  not  deal  with  a  set  of  things 
that  we  might  fancy  arranged  in  a  row,  like 
a  row  of  blocks,  beginning  here  and  ending 
there.  Science  is  interested  only  when  the 
row,  if  it  begins,  never  ends.  Consider  a 
curve.  You  can  not  exhaust  its  points  by 
naming  one  after  another  of  them.  That 
is  just  why  science  is  interested.  It  deals 
with  the  totality  of  the  points  by  dealing 
with  the  curve,  that  is  with  the  law — a 
definite  thing — defining  the  totality,  which 
is,  therefore,  also  definite.  Do  you  think 
geometry  would  exist  if  the  points  of  space 
could  be  counted  like  a  heap  of  marbles?  If 
it  did,  it  would  be  trivial.  So  much  by  way 
of  reassuring  those  timid  persons  who,  pri- 
marily children  of  sense  and  imagination,  are 


68          THE  NEW  INFINITE  AND 

filled  with  doubt  and  trepidation  when  asked 
to  pass  upward  from  their  accustomed  atmos- 
phere into  the  ether  of  pure  thought. 

Let  us  now  resume  the  advance.  Compare 
the  totality  (  W)  of  integers  with  the  totality 
(P)  of  even  integers.  Let  us  agree  to  pair 
each  integer  of  (  W)  with  the  one  below  it  in 
(P).  In  this  way  a  one-to-one  correspond- 
ence is  set  up  between  the  integers  in  (W) 
and  those  in  (P),  a  result  that  we  may  indi- 
cate by  the  following  sequence  of  pairs : 

(T)     1,2;2,4;3,6; ;n,2n;  .  .  . 

Observe  that  the  pairing  is  no  creeping 
performance  that  never  gets  performed — 
ever  going  on  and  never  finishing;  neither  is 
it  a  lightning-swift  process,  for  this  were 
as  helpless  before  the  task  of  pairing  the 
totalities  step  by  step  as  would  be  the  pace 
of  a  snail:  an  endless  course  can  not  be  run 
through  by  merely  going  fast.  No,  the 
pairing  is  an  instantaneous  deed  of  law, 
wrought  without  lapse  of  time.  The  law  is: 
each  number  shall  go  with  its  double.  To 
choose  the  law  is  to  say :  Let  the  pairing  be 
done;  and — it  is  done.  To  contemplate  the 
deed  requires  time ;  but  the  doing  of  it,  none. 


THE  OLD  THEOLOGY  59 

There  is  possible  a  yet  deeper  view  of  the 
matter.  I  mean  the  static  view.  We  may 
say,  that  is, — and  this  is  correct — ,  that  the 
integers  as  elements  of  the  existing  world  of 
ideas  already  and  always  stand  at  once  in 
all  sorts  of  interrelations  of  which  it  is  the 
nature  of  integers  to  admit,  among  such 
relations  being  that  indicated  by  (T).  In 
this  view,  the  pairing  is  not  a  process  of 
associating  an  integer  with  its  double,  then 
another  with  its  double,  and  so  on,  thus 
establishing  progressively,  so  to  speak,  the 
relation  (T).  It  is  not  that;  it  is  simply 
a  single  act  of  will  choosing  out  a  certain 
eternal  relation  from  among  hosts  of  rela- 
tions also  eternal.  Whichever  view  of  the 
matter  be  taken — and  either  is  admissible — 
it  is  clear  that  a  one-to-one  relation  does 
subsist  between  the  elements  in  (TF)  and  the 
elements  in  (P).  The  two  totalities  are 
therefore  equally  rich  in  elements:  the 
number  of  integers  in  the  one  is  the  same 
as  the  number  of  those  in  the  other.  But 
every  integer  in  (P)  is  an  integer  in  (TV), 
while  (W)  has  integers  that  are  not  in  (P). 
Hence  (P)  is  a  part  and  (W)  the  whole; 
and  so  ( W)  contains  an  infinitude  of  inte- 


60          THE  NEW  INFINITE  AND 

gers;  and  the  like  is  true  of  (P),  for  what- 
ever matches  an  infinite,  in  the  way  now 
repeatedly  exemplified,  is,  of  course,  itself 
infinite — indeed,  infinite  of  the  same  rank. 

It  is  needless,  I  trust,  to  cite  here  further 
examples.  "These  slight  footprints  suffice 
to  enable  a  keen-searching  mind  to  find 
out" — not  "aZZ  the  rest",  as  the  maddened 
poet  sang — but  more  and  more.  For,  to 
eyes  once  opened,  the  brood  of  the  infinite 
is  everywhere.  The  light  of  the  great  con- 
cept shines  in  every  aspect  of  being.  The 
reader  is  now  aware  that  this  our  world  is 
a  world  that  presents  two  great  types  of 
wholes,  or  manifolds,  of  thinkable  realities — 
manifolds  that  are  finite  and  manifolds  that 
are  infinite.  He  is  now  aware  that  each  of 
the  latter  is  characterized  by  the  marvelous 
fact  that  it  is  a  whole  containing  a  part 
(countless  parts  indeed)  matching  the  whole 
perfectly,  as  we  have  seen,  in  elemental 
wealth,  in  richness  of  content,  in  dignity  of 
structure.  The  principle  of  discrimination 
is  very  simple — so  simple  indeed  as  to  have 
eluded  the  eye  of  thought  for  thousands  of 
years — for  the  doctrine  is  very  modern,  a 
faint  first  glimmer  of  it  appearing  in  a  work 


THE  OLD  THEOLOGY  61 

of  Galileo  and  a  little  later  in  a  hint  of 
Pascal  but  not  again,  it  seems,  for  two  cen- 
turies. By  it  the  universe  of  thinkable 
reality,  as  we  now  see,  is  riven  asunder,  not 
spatially  indeed  but  logically.  The  two 
grand  divisions — the  realm  of  the  finite  and 
the  realm  of  the  infinite — ,  which  are  wonder- 
fully interlocked,  together  constitute  a  dual 
world  answering  to  our  dual  life,  the  life  of 
action  and  the  life  of  thought.  The  realm 
of  finite  things  is  the  domain  of  action,  of 
Practical  Life :  it  contains  no  multitudes  but 
man  may  count  them — the  coins  in  the  coffer, 
the  cattle  in  the  field,  the  deeds  of  a  hero,  the 
years  of  an  empire;  any  series  in  it  begins 
and  ends;  no  totality  or  whole  found  there 
is  matched  by  one  of  its  parts :  the  world  of 
finite  things  is  an  island-world  suspent  in  a 
sea.  And  what  is  the  immersing  sea?  It  is 
the  realm  of  infinite  things — an  ocean  with- 
out bottom  or  surface  or  shore.  It  contains 
no  totalities  but  such  as  are  law-defined, 
never  a  whole  of  any  kind  that  has  not 
countless  parts  each  matching  it  perfectly 
in  respect  of  number  of  elements,  coequal 
with  it  in  Machtigkeit  as  it  is  called,  in 
potence  or  power,  in  complexity  of  structure, 


62          THE  NEW  INFINITE  AND 

in  dignity  and  wealth  of  reality.  This  is 
not  the  domain  of  Practical  Life,  though  it 
penetrates  the  latter  domain,  intersects  it 
in  numberless  ways,  surrounds  it,  contains 
it  in  a  sense:  for  a  series  that  terminates  is 
but  part  of  one  that  does  not ;  every  ensemble 
that  admits  of  tabulation  is  a  fragment  of 
one  than  can  not  be  fully  represented  by 
tabulation  but  only  by  a  law;  every  whole 
that  is  an  overmatch  for  its  every  part 
belongs  to  some  vaster  whole  owning  parts 
with  respect  to  which  it  is  not  an  over- 
match; every  finite  manifold  is  a  sub- 
collection  of  an  infinite  one.  No,  the  realm 
of  infinite  totalities,  though  embracing,  in 
the  sense  explained,  the  domain  of  Practical 
Life,  is  not  that  domain;  it  is  the  domain  of 
Reason,  the  province  of  Thought,  the  realm 
of  Science;  for,  as  Poincare  has  acutely 
pointed  out,  there  can  be  no  science,  prop- 
erly speaking,  of  a  finite  subject-matter. 

Very  well,  one  may  wish  to  say,  I  grant 
what  you  have  said,  but  what  of  it?  Where, 
pray,  is  Deity?  I  ask  for  bread;  you  give 
me  a  stone.  I  ask  for  a  vision  of  God;  you 
invite  me  to  thread  endless  mazes  of  mathe- 
matics ;  you  invite  me  to  contemplate  vast 


THE  OLD  THEOLOGY  68 

and    dazzling    splendors    of    Number    and 
Space.    What  does  it  all  avail? 

"  I  heap  up  numbers  enormous, 
Mountains  of  millions  extend, 
Piling  time  upon  time, 
World  on  world  without  end, 
But  when  from  the  awful  height 
I  would  a  vision  of  Thee  behold: 
The  total  sum  of  number's  Might, 
Tho'  multiplied  a  millionfold, 
Is  yet  no  part  of  Thee." 

The  protest  is  easy  to  understand,  it  is 
temperamental.  May  I  reply,  by  way  of  a 
reminder,  that  I  have  promised  no  "vision" 
of  God?  I  am  dealing  with  the  hypothesis 
God.  My  aim  is  to  throw  some  light  on  the 
meaning  of  its  mighty  terms.  Chief  among 
these  is  the  theological  Infinite.  In  pur- 
suance of  the  aim  I  have  been  here  trying 
to  clarify  the  meaning  of  scientific  infini- 
tudes, of  which  the  Infinite  of  theology  is 
the  supernal  ideal  or  limit.  None  but  the 
infinite,  it  is  said,  can  comprehend  the  infi- 
nite. How  familiar  are  the  words!  How 
often  have  they  been  solemnly  pronounced 
in  courts  of  philosophy  and  sunken  in  the 
soul  like  a  leaden  decree  of  fate !  But  are 
they  not  true?  "Comprehend"  here  means, 


64          THE  NEW  INFINITE  AND 

of  course,  comprehend  rationally,  it  signifies 
to  understand  as  ideas  are  understood.  Said 
of  the  infinites  of  science,  the  words  are, 
then,  true.  Said,  however,  of  the  theological 
Infinite,  they  are  neither  true  nor  false ;  they 
are  meaningless,  for  the  theological  Infinite 
is,  as  already  said  and  as  I  hope  we  are 
beginning  to  see,  a  superrational  ideal,  and 
to  talk  of  comprehending  or  not  comprehend- 
ing such  an  ideal  as  we  talk  of  understanding 
ideas  is  not  to  utter  what  is  true  or  false 
but  what  is  void  of  meaning.  If,  however, 
'comprehend  the  Infinite'  be — contrary  to 
usage — taken  to  mean,  not  comprehend  it  as 
an  idea,  which  it  is  not,  but  to  signify  having 
or  gaining  that  kind  of  sense  of  what  it 
means  which  comes  from  regarding  it  as  the 
ideal  or  limit  of  infinites  that  we  can  com- 
prehend as  ideas,  then  the  old  maxim, 
applied  to  the  theological  Infinite,  is  false, 
for  we  can  win  the  mentioned  sense  in  the 
mentioned  way,  and  we  do  not,  I  believe, 
regard  ourselves  as  superrational  ideals. 
But  have  we  not  involved  ourselves  in  con- 
tradiction? For  how  can  we  gain  the  men- 
tioned sense  in  the  mentioned  way,  seeing 
that  this  way  requires  ordinary,  or  logical, 


THE  OLD  THEOLOGY  65 

comprehension  of  infinites,  and  seeing  that, 
regarding  these  infinites  of  science,  we  have 
admitted  that  none  but  the  infinite  can 
comprehend  the  infinite?  The  answer  is  no, 
there  is  no  contradiction,  for  we  are  our- 
selves infinite  in  the  scientific  meaning  of  the 
term,  where  by  "we"  I  mean  the  common- 
wealth of  ideas  over  which  your  mind  or  mine 
can  range.  That  this  is  true  we  know  at 
length,  thanks  to  mathesis,  and  we  know  it, 
not  merely  as  an  intimation  or  intuitive 
apprehension,  but  as  a  proved  proposition 
of  science.  We  know,  that  is,  as  Richard 
Dedekind  has  rigorously  demonstrated,  that 
the  world  of  man's  ideas  as  ideas — the  human 
Gedankenwelt  as  the  author  calls  it — is  an 
infinite  manifold.  Shorn  of  contest  and 
other  non-essentials,  the  proof  may  be  ren- 
dered in  a  line.  Some  readers  will  not 
require  it.  A  friend  tells  me  that  he  does 
not  understand  the  proof  and  does  not  need 
it.  I  may  add  that  he  is  a  man  of  extraor- 
dinary spiritual  sensibility.  Intuition,  how- 
ever, precious  as  it  is,  is  often  wrong,  and 
I  give  the  proof  for  the  comfort  of  those  who 
think  it  important  to  submit  their  intuitions, 


66          THE  NEW  INFINITE  AND 

when  it  is  possible,  to  the  rigors  of  logical 
demonstration. 

Denote  by  G  the  Gedankenwelt — the  world 
of  ideas;  by  /  any  idea  therein,  as  that  of  a 
meal,  a  song,  a  deed  of  charity,  a  bargain, 
a  moon  beam,  a  diamond,  a  birth,  a  death; 
by  7i  the  idea  of  7,  for  plainly  the  idea  that 
a  given  idea  is  an  idea  is  another  idea;  by 
7 2  the  idea  of  A;  and,  generally,  by  I^-\-\  the 
idea  of  7n.  As  any  thought  may  itself  be 
object  of  another  thought,  as  this  one  may 
be  object  of  a  third  different  from  the  former 
two,  and  so  on  forever,  it  is  seen  that  7n+i 
can  never  fail,  however  large  n  may  come 
to  be,  and  so  we  have  the  two  totalities : 

(T)    I,  Ji,  /2,  .  .  .  .  ,  /n,  Jn+i,  .  .  .  .  ; 

(T)     h,  h,  Is,  .    .    .    .  ,  /n+l,  /n+2 ; 

the  latter  is  a  part  of  the  former;  each  of 
them  is  a  part  of  G.  Now  pair  the  two 
totalities  as  in  the  following  scheme: 

/,  /i ;  /i,  /2 ;  /2,  /s  ; .  .  .  .  ;  /n,  /n+i ;  /n+i,  /n+2  ;....; 

each  thing  in  (T)  being  thus  associated  with 
the  thing  below  it  in  (T").  At  once  it  is 
seen  that  the  whole  totality  (T)  is  com- 
pletely matched  in  one-to-one  fashion  by  its 


THE  OLD  THEOLOGY  67 

part  (T');  whence  it  follows  that  (T)  is 
infinite;  that  (T')  is  infinite;  and,  a  fortiori, 
that  their  common  container,  G,  the  Gedan- 
Tcenwelt,  is  infinite. 

In  this  simple  demonstration,  so  free  from 
pomp,  and  in  its  conclusion,  so  significant 
for  a  right  conception  of  man,  there  is  large 
gain  for  rational  theology,  if  indeed  we  may 
hope  that  professional  theologians  will  one 
day  be  moved  to  avail  themselves  of  such 
considerations.  It  is  no  small  gain  to  vindi- 
cate by  logic  a  great  intuition  of  the  soul: 
it  is  no  small  thing  to  know,  not  merely  at 
times  to  feel,  that  our  faculties  are  framed 
to  comprehend,  scientifically,  infinite  elements 
in  the  architecture  of  the  world.  For  in  the 
presence  of  such  knowledge,  the  terrors  of 
Naturalism  dwindle  and  vanish.  Kant's 
exclamation  that  "modern  astronomy  has 
annihilated  my  own  importance"  ceases  to 
have  significance  when  once  we  know  that 
with  countless  infinitudes  encountered  in 
time  and  space  our  faculties  are  competent 
to  deal, 

"  Times  unending 
Comprehending, 
Space  and  worlds  of  worlds  transcending." 


68          THE  NEW  INFINITE  AND 

We  desire  no  instauration  of  the  shallow 
and  timid  humanism  that  derived  its  estimate 
of  man  from  a  geocentric  theory  of  the  uni- 
verse, cried  alarm  at  the  crumbling  of  a 
Mosaic  cosmogony  and  shudders  still  at  the 
shrinking  of  the  earth  to  a  pebble  in  the 
cosmic  perspectives  opened  to  the  view  by 
modern  science.  Bigness  does  not  daunt 
Mathesis;  she  seeks  it;  vastness  is  the  aether 
that  sustains  her  wing.  In  her  modern  doc- 
trine of  infinite  manifolds,  of  which  I  am  here 
trying  to  give  a  rather  slight  indeed  but  hint- 
ful  sketch,  she  has  extended  the  dominion 
of  logic  far  beyond  the  utmost  borders  of 
finite  things  out  into  the  realm  of  transfinite 
reality.  And  when,  if  ever,  theology  learns 
to  follow  thither,  when  if  ever,  she  acquaints 
herself  with  the  procedures  of  science  there 
and  learns  to  contemplate  the  innumerable 
infinitudes  that  science  can  understand,  she 
will  find  that  the  hierarchy  they  constitute 
is  a  ladder  for  her,  an  endless  ladder  by 
which  she  may  ascend  higher  and  higher  into 
a  better  and  better  sense  of  what  she  ought 
to  mean  by  her  own  Infinitude,  which  at  once 
o'ertops  and  includes  them  all. 

At  this  point  the  reader  may  naturally 


THE  OLD  THEOLOGY  69 

desire  to  enter  the  discussion  and  have  a 
share  in  shaping  its  course.  I  have,  he  may 
wish  to  say,  now  acquired  a  pretty  clear  con- 
ception of  what  science  means  by  an  infinite 
manifold;  I  have  grasped  the  abstract  idea 
and  have  seen  it  realized  and  illustrated  in 
a  variety  of  concrete  examples;  I  am  now 
prepared  to  find  other  examples  for  myself, 
for  I  am  beginning  to  see  that  such  multi- 
plicities compose  the  intelligible  portion  of 
the  embracing  world,  that  they  are  literally 
omnipresent,  that  even  in  the  surface  of 
common  life  and  common  thought  they  gleam 
here,  there  and  yonder  like  shining  bassets 
of  gold.  But  I  do  not  see,  he  may  say,  that 
they  are  not  of  a  single  type;  I  have  not 
glimpsed  the  ladder;  I  am  far  from  seeing 
that,  in  respect  of  dignity,  they  dispose 
themselves  rank  above  rank  in  a  hierarchy 
without  a  rank  supreme. 

These  words  of  the  reader  imply  a  legiti- 
mate demand,  which  must  now  be  met — met, 
that  is,  in  so  far  as  circumstances  will  allow, 
for  the  matter  is  pretty  subtle,  involving 
some  technical  considerations  known  only  to 
mathematicians,  and  does  not  admit  of  pub- 
lic presentation  in  a  line.  It  is  possible, 


70          THE  NEW  INFINITE  AND 

however,  without  too  many  words,  so  to 
delineate  the  matter  as  to  give  what  is  suffi- 
cient— a  realizing  sense  of  its  truth.  For 
this  purpose  I  must  ask  the  reader  to  look 
again  at  the  infinite  manifold  denoted  above 
by  (  W).  It  is  a  homely  affair.  How  dreary 
it  looks  and  commonplace.  But  let  us  not 
be  disheartened.  We  are  going  to  see  that 
it  has  beautiful  aspects  not  yet  disclosed,  a 
dignity  and  character  to  quicken  the  pulse 
of  our  thought  and  win  our  admiration;  for 
we  are  going  to  see  that  it  belongs  to  an 
immense  family  of  similar  manifolds,  many 
of  them  of  singular  beauty, — a  countless  host 
of  them — ,  and  that  this  homely  one  has  the 
distinction  and  honor  to  represent  the  type. 
Of  this  family  many  members  differ  so  much 
in  appearance  as  to  have  concealed  for  thou- 
sands of  years  the  deep  similitude  that  makes 
them  kin.  Why  is  it,  I  wonder,  that  things 
are  not  what  they  seem?  Is  it  because,  if 
they  were,  life  would  be  void  of  its  finest 
interest — the  zest  of  research,  the  joy  of  dis- 
covery, the  surprise  and  delight  of  detecting 
the  hid?  Perhaps  so,  but  let  the  question 
pass,  and  attend  very  sharply  to  what  is  now 
to  be  said.  Any  ensemble  or  manifold  of 


THE  OLD  THEOLOGY  71 

elements  such  that  a  one-to-one  correspond- 
ence can  be  set  up  between  them  and  the 
numbers  composing  the  manifold  (  W)  is  said 
to  belong  to  the  type  of  infinite  manifolds 
represented  by  (  W) .  This  type  has  a  beau- 
tiful designation — it  is  called  the  Denumer- 
able  Type:  the  manifolds  belonging  to  it  are 
called  denumerable  infinities  or  denumerably 
infinite  manifolds  or  classes.  Their  name  is 
legion.  One  of  them,  as  we  have  seen,  is  the 
manifold  of  even  integers,  above  denoted  by 
(P);  obviously  another  is  the  ensemble  of 
odd  integers;  another  may  be  got  by  taking 
from  (  W)  for  elements,  say  the  number  one, 
then  a  million,  then  a  million  million,  and  so 
on ;  it  being  thus  evident  that  (  W)  contains 
countless  denumerably  infinite  parts.  But 
let  us  go  outside  of  (W).  Consider  any 
straight  line  L  running  endlessly  up  and 
down — zenith-ward  and  nadir-ward — pierc- 
ing or  passing  the  stars,  like  a  thread 
stretched  through  the  universe  of  space. 
Consider  the  ensemble  of  miles  it  contains. 
On  it  choose  a  starting  point  0.  Conceive 
as  marked  by  1  the  end  of  the  first  upward 
mile,  by  2  the  end  of  the  first  downward  mile, 
by  3  the  end  of  the  next  upward  mile,  by  4 


72          THE  NEW  INFINITE  AND 

the  end  of  the  next  downward  mile,  and  so 
on  and  on.  Will  the  integers  fail?  No,  you 
say.  Will  thread  length  fail?  Again  you 
say  no.  So,  then,  the  law  of  correlation 
holds,  the  required  correspondence  is  estab- 
lished between  the  mile  posts  of  L  and  the 
integers  of  (  W) .  Obviously  the  same  would 
be  the  case  if  we  chose  any  other  unit  of 
length.  Accordingly,  we  see  that  the  en- 
semble of  unit  lengths  composing  a  thread 
or  course  that  traverses  endlessly  the 
abysses  of  Space  is  an  infinite  manifold  of 
the  denumerable  type.  To  the  same  type 
of  infinitude  plainly  belongs  the  aggregate 
of  minutes  or  hours  or  centuries  in  the 
stretch  or  course  of  Time  conceived  as  run- 
ning eternally  backward  and  eternally  for- 
ward. Well  might  the  apostle  exclaim  that 
one  day  is  with  the  Lord  as  a  thousand  years, 
and  a  thousand  years  as  a  day.  No  wonder 
Lucretius  could  say  that,  however  many 
years  you  may  prolong  your  life,  you  can 
not  diminish  by  a  single  jot  the  length  of 
time  you  will  be  dead.  Without  knowing  it, 
these  men  were  thinking  in  terms  of  denu- 
merable infinitudes.  They  did  not  indeed 
understand  the  matter  scientifically,  but  they 


THE  OLD  THEOLOGY  78 

felt  it,  and  in  their  utterance  is  the  throb 
of  its  mighty  power.  I  wish  now  to  present 
what  is — if  one  will  but  ponder  it  till  it  is 
clearly  seen  in  the  light  of  meditation — 
perhaps  the  most  impressive,  certainly  the 
most  astonishing,  known  example  of  the  type 
of  infinity  here  in  question.  Consider  the 
totality  of  rational  fractions,  of  those  frac- 
tions, that  is,  whose  terms  (numerators  and 
denominators)  are  integers,  or  whole  num- 
bers. Take  any  two  integers,  say  3  and  4, 
and  reflect  a  little  upon  the  multitude  of 
fractions  that  lie  between,  being  greater, 
that  is,  than  3  and  less  than  4.  Take  any 
two  of  these;  between  them  there  is  another 
and  another;  between  these,  another  and 
another ;  and  so  on  forever.  How  thickly 
they  are  crowded  together !  More  numerous 
than  the  sands  of  the  sea,  than  the  drops  in 
the  ocean,  for  these  sands  or  drops,  if 
arranged  in  a  row,  would  not  go  on  forever. 
In  the  interval  between  any  two  consecutive 
integers  stands,  then,  a  countless  crowd  of 
fractions.  Do  but  reflect  and  reflect  again 
upon  the  amazing  multitude:  an  infinite  host 
in  each  interval  of  an  infinite  host  of  inter- 
vals. Surely  we  have  here — have  we  not? — 


74          THE  NEW  INFINITE  AND 

in  this  infinity  of  infinities  of  rational  frac- 
tions an  overmatch  for  the  mere  ensemble 
(W)  of  integers.  Undoubtedly  it  seems  so. 
But  it  is  seeming  only:  the  appearance 
deceives.  The  imposing  array  of  all  the 
fractions,  as  soon  we  shall  see,  belongs  to 
the  denumerable  type  of  infinitude.  Nay,  we 
may  even  throw  all  the  integers  and  all  the 
rational  fractions  together,  and  then  show 
that  the  new  multitude  is,  in  respect  of 
multiplicity,  perfectly  matched  by  the  array 
of  integers  alone,  notwithstanding  these  seem 
in  comparison  so  few  and  scarce.  Let  us 
prove  this  astounding  fact,  for  it  is  but  a 
sample — a  model,  if  you  please,  or  pattern — 
of  surprising  relationships  literally  saturat- 
ing the  subject-matter  of  theology  and  there 
awaiting  disclosure  for  the  enlightenment 
and  edification  of  man.  The  argument  is 
easy  to  follow.  Take  a  fraction  at  random, 
say  % ;  note  that  the  sum  of  its  terms — its 
term-sum — is  an  integer,  in  this  case  5 ;  note 
that  there  are  other  fractions  having  the 
same  term-sum;  arranged  in  the  order  of 
increasing  numerators,  they  are:  %,  %, 
%,  %.  Any  other  integer  will  similarly  give 


THE  OLD  THEOLOGY  75 

rise  to  a  set  of  fractions,  which  we  may 
arrange  in  similar  order.  Thus  the  term- 
sum  2  gives  (a)  :  /•!.  The  term-sum  3  gives 
(b) :  Yz,  %.  The  term-sum  4  gives  (c)  : 
%»  %•>  %.  The  term-sum  5  yields  (d) : 
%»  %>  %>  %•  The  term-sum  6  furnishes 
(e):  Vst  %,  %,  %,  %•  And  so  on  forever. 
Observe  that  this  procedure  is  one  that 
sooner  or  later  presents  us  with  any  fraction 
whatever  that  we  may  designate.  Whole 
numbers  appear  among  the  fractions,  as 
%  or  %,  for  example,  and  a  same  integer  is 
repeated,  as  %  and  %,  for  example;  and  a 
same  fraction  appears  repeatedly,  as  %,  %, 
for  example.  We  agree,  however,  to  take 
each  but  once  in  the  matching  process  now 
to  follow.  Bear  in  mind  what  we  are  to  show : 
it  is  that  the  integers  of  (  W) ,  taken  alone, 
perfectly  match,  in  one-to-one  fashion,  all 
the  rational  fractions  and  all  the  integers 
taken  together.  The  correlation  proceeds  as 
follows:  pair  1  of  (W)  with  •/•!  of  (a)  ;  next 
pair  2  and  3  of  (W)  respectively  with  % 
and  %  of  (b)  ;  next  pair  4  and  5  of  (W) 
respectively  with  %  and  %  of  (c),  omitting 
%  as  a  repetition  of  %  already  paired;  and 


76          THE  NEW  INFINITE  AND 

so  on  and  on.     We  thus  get  the  following 
scheme  of  one-to-one  association: 

1.  M;  2,  %,  3,  %;  4,  %,  5,  %;  6,  %,  7,  %,  8,  %,  9,  ft; 


Observe  that  the  law  of  procedure  matches 
each  integer  of  (TF)  with  some  definite 
fraction  or  whole  number,  and  each  fraction 
or  whole  number  in  the  grand  totality  of 
fractions  and  whole  numbers  with  a  definite 
integer  of  (W).  The  result,  then,  is  this: 
that  grand  totality,  embracing,  as  we  saw, 
an  infinitude  of  infinitudes  of  things,  so  far 
surpassing,  it  seemed,  in  elemental  wealth  the 
manifold  of  integers,  is  nevertheless  perfectly 
matched  by  it  in  that  regard,  owns  precisely 
the  same  Machtigkeit,  and  is  thus  only  an 
exceptionally  impressive  member  of  the  great 
family  or  type  of  Denumerable  Infinity. 

On  discovering  results  so  astonishing  as 
the  one  I  have  just  now  presented,  it  is  little 
wonder  that  mathematical  students  of  the 
subject  suspected  for  a  time  that  possibly 
all  thinkable  or  discoverable  infinitudes  would 
be  found  upon  examination  to  be  of  one 
family,  of  one  and  the  same  type — the  denu- 
merable  type — ,  however  much  one  infinity 


THE  OLD  THEOLOGY  77 

might  seem  to  surpass  another  in  wealth  of 
elements.  But  the  suspicion  was  short-lived : 
it  was  soon  discovered  that  everywhere  round 
about  us  there  are  innumerable  infinitudes  of 
higher  type — infinitudes,  that  is,  such  that 
you  can  no  more  exhaust  the  wealth  of  one 
of  them  by  removing  from  it  a  denumerable 
infinity  of  its  elements  than  you  can  exhaust 
a  denumerable  infinitude  by  taking  from  it  a 
finite  collection  however  large.  Indeed  it  is 
now  well  known  that  the  denumerable  type 
is  the  lowest  type  of  infinite  manifolds  and 
that  above  it,  as  I  have  said,  there  rise  in  the 
world  of  thought  an  endless  scale  of  types. 
I  regret  the  necessity  in  this  connection  of 
having  to  request  the  reader,  if  he  be  not  a 
mathematician,  to  accept  a  few  mathematical 
facts  or  propositions  on  the  authority  of  my 
report,  for  to  prove  all  of  them  here  would 
both  expand  this  volume  beyond  desirable 
limits  and  include  in  it  argumentation  of  a 
kind  too  technical  for  the  general  reader, 
whatever  his  abilities  or  attainments  in  other 
fields.  The  reader  knows  that  besides  the 
rational  numbers,  above  considered,  there 
exist  what  we  call,  somewhat  unhappily  but 
for  good  historical  reasons,  irrational  num- 


78          THE  NEW  INFINITE  AND 

bers.  He  knows  that  these  are  such  as  V2, 
the  number  denoted  by  IT — ratio  of  the  cir- 
cumference to  the  diameter  of  a  circle — ,  the 
number  denoted  by  e — base  of  the  Naperian 
system  of  logarithms — and  many  others 
equally  familiar.  He  probably  does  not 
know — what  is  nevertheless  true — that  the 
irrationals  are,  unlike  the  rationals,  not 
denumerable;  they  are  too  numerous  for 
that — a  fact  that  mathematicians  have 
rigorously  demonstrated;  the  irrationals 
constitute  an  infinite  manifold  of  higher  rank 
or  type  than  the  denumerable  type;  if  from 
the  totality  of  irrationals  we  take  away  a 
denumerable  infinitude  of  them  there  will 
always  remain  infinitely  more  irrationals 
than  we  have  taken  away.  Nay,  this  will  be 
true  if  we  take  away,  not  merely  one  denu- 
merable infinitude  of  them,  but  another  such, 
then  another,  and  so  on  endlessly,  thus 
removing  a  denumerable  infinitude  of  denu- 
merable infinitudes  !  The  original  ensemble 
remains  absolutely  undiminished — its  wealth 
of  elements,  its  Mdchtigkeit,  or  "power",  its 
dignity,  its  rank  or  type,  is  the  same  as 
before  the  great  removal  or  decimation. 
This  wonderful  type  of  infinitude  has,  like 


THE  OLD  THEOLOGY  79 

the  other  type  considered,  a  fine  name:  it  is 
called  the  Type  of  the  Continuum;  it  is  so 
called  because  the  totality  of  points  in  a 
continuous  line  of  any  length,  however  short, 
is  a  familiar  example  of  an  infinite  manifold 
belonging  to  the  type  in  question.  "What 
is  it  that  you  say?"  may  interject  a  reader. 
"Do  you  mean  to  tell  me  the  ensemble  of 
points  in  a  little  short  line  or  the  ensemble 
of  instants  in  a  little  stretch  of  time  would 
not  be  exhausted  if  we  could  take  away  from 
it  a  denumerable  infinitude  of  points  in  the 
one  case  or  of  instants  in  the  other?"  The 
answer  is,  I  do:  such  a  taking  away  would 
not  only  not  exhaust  the  ensemble  but  it 
would  not  even  diminish  its  wealth  of  ele- 
ments— no,  not  if  the  great  subtraction  were 
repeated  a  billion  times  in  each  second  in 
an  endless  succession  of  seconds !  Assuming 
that  this  is  mathematically  sound,  which  it 
is,  is  it  unreasonable  to  say,  as  I  do  say,  that 
our  amiable  theological  friends  and  guides, 
in  preparing  to  instruct  regarding  the  theo- 
logical Infinite,  which  is  over  and  above  all 
other  infinitudes,  would  do  well  to  gain  some 
insight  into  the  wonders  of  those  that  are 
below?  Is  it  unreasonable  to  contend  that 


80          THE  NEW  INFINITE  AND 

a  course  of  lectures  in  this  matter  ought  to 
be  regularly  provided  in  our  theological 
seminaries?  Personally  I  have  no  doubt  at 
all  that  a  competent  student  of  theology 
could  be,  not  only  much  informed,  but 
thrilled,  joyed,  and  inspired,  by  the  marvels 
of  insight  and  perspective  that  such  a  course 
could  open  to  his  gaze.  That,  however,  by 
the  way.  Members  of  the  great  family  of 
the  Continuum  type  of  infinity  are  omni- 
present in  our  world.  One  of  them  is  the 
manifold  of  rational  and  irrational  numbers 
taken  together;  another  is  the  collection  of 
instants  in  a  second  or  a  thousand  years; 
another  is  the  ensemble  of  points  in  a  sphere 
or  in  the  universe  of  space;  another  is  the 
ensemble  of  angles  among  the  lines  of  a  sheaf 
or  the  ensemble  of  the  lines  themselves; 
another  is  the  pencil  of  planes  having  a  line 
in  common,  or  the  collection  of  spheres 
centered  at  a  point,  or  the  totality  of  rela- 
tions between  points  of  time  and  states  of 
the  world,  or  the  aggregate  of  possible 
motions,  or  the  group  of  possible  poses  of 
a  tiger  or  a  statue,  or  the  multitude  of 
simple  equations  showing  how  two  variables 
may  change  together,  or  the  multitude  of 


THE  OLD  THEOLOGY  81 

spaces  like  ours  that  coexist  in  a  space  of 
higher  dimensionality;  and  so  on  and  on 
endlessly  and  forever.  Is  the  Continuum 
type  the  next  above  the  Denumerable  type? 
Probably  so  but  no  one  knows.  Whoever 
answers  the  question  will  thereby  immor- 
talize his  name. 

Shall  we  proceed  to  infinite  types  that  are 
superior  to  that  of  the  Continuum  in  respect 
of  dignity  or  rank?  It  were  possible  so  to 
do  but  the  way  is  steep.  I  fear  to  weary  the 
reader  by  too  much  demonstration  of  what 
he  is  now,  I  venture  to  hope,  prepared  to 
assume.  For  as  we  go  up  from  level  to  level 
in  the  ever  ascending  scale  of  more  and  more 
embracing  infinitudes,  the  thought  becomes 
abstracter  and  abstracter,  the  heights 
dizzier  and  dizzier.  At  least  for  the  present 
we  have  perhaps  climbed  enough.  At 
another  time,  when  our  lungs  have  become 
accustomed  to  the  tenuous  air,  the  ascent 
may  be  resumed.  Here  and  now  it  is  suffi- 
cient to  know  that  the  hierarchy  exists,  that 
each  rank  includes  all  ranks  below  it,  and 
that,  taken  together,  these  ranks  above 
ranks  of  Infinitude,  amenable  to  the  ways  of 
Reason,  constitute,  as  we  have  said,  an  end- 


82          THE  NEW  INFINITE  AND 

less  ladder,  an  ever  rising  scale,  along  which 
the  subtle  process  of  Idealization,  with  the 
velocity  of  spirit,  proceeds  upward  forever, 
attaining  never  a  summit,  for  there  is  no 
summit,  but  intimating,  indicating,  and  ever 
approximating,  an  outlying  Limit,  supernal, 
above  all  ranks,  embracing  all,  reflecting  the 
glory  of  all — the  Infinite  of  theology. 

The  foregoing  sketch  has,  I  trust,  made  it 
fairly  clear  in  a  general  way  that  in  the 
study  of  the  infinites  of  science,  which  are 
infinite  ideas,  and  not  elsewhere,  there  is  a 
scientific  way  to  the  meaning  of  theology's 
Infinite,  which  is,  not  an  infinite  idea,  but 
something  more  supreme — an  infinite  ideal. 
In  "a  general  way,"  I  have  said,  for  the 
considerations  adduced  have  been,  in  the 
main,  pretty  broad  and  general.  I  wish  now 
in  approaching  the  end — for  this  writing 
must  terminate — to  descend  to  particulars 
and  to  show  by  some  concrete  examples 
how  the  study  in  question  can  render  the 
service  claimed. 

As  every  one  knows,  the  indictment  that 
men  of  rationalistic  temper,  including  for  the 
most  part  scientific  men,  have  brought 
against  theology,  is  not  the  same  as  their 


THE  OLD  THEOLOGY  83 

objection  to  religion.  It  is  very  far  from 
the  same.  Their  indictment  of  theology 
charges  that  theology  is  not  coherent,  that 
it  is  replete  with  internal  contradictions,  that 
it  thus  fails  to  meet  the  rightful  demand  of 
intellect  for  harmony,  and  so  fails  to  meet 
the  standard  essential  alike  to  science  and  to 
art.  The  indictment  is  fatal  unless  the 
alleged  contradictions,  familiar  to  all,  can 
be  purged  away  or  else  transcended.  Broadly 
speaking  they  are  of  two  kinds — foreign  and 
domestic — contradictions,  that  is,  arising 
from  theology's  use  of  assumptions  or  postu- 
lates that,  however  available  elsewhere,  are 
entirely  outside  theology's  proper  domain, 
and  contradictions  that  do  not  arise  from 
imported  postulates  but  present  themselves 
properly  in  theology's  subject-matter  as  seen 
from  interior  but  inadequate  or  fragmentary 
points  of  view.  Contradictions  of  the  for- 
eign variety  may,  I  think,  be  gradually 
purged  away  by  ridding  theology  of  imported 
postulates ;  contradictions  of  the  domestic 
kind  may,  I  am  equally  confident,  be  tran- 
scended more  and  more  by  seeking  view- 
points more  and  more  commanding. 

I  wish  to  indicate  what  appears  to  me  to 


84          THE  NEW  INFINITE  AND 

be  the  right  manner  of  dealing  with  these 
two  comprehensive  varieties  of  theological 
contradiction  or  difficulty.  And  first  a  word 
respecting  the  foreign  kind.  These  are  like 
the  contradictions  that  would  defeat  the 
ends  of  justice  if,  in  the  trial  of  a  case  at 
law,  it  were  assumed  and  held  throughout 
that  all  witnesses  are  honest  or  that  none 
can  be  mistaken;  or  like  the  hopeless  con- 
fusion that  would  result  to  the  science  of 
hydraulics,  did  the  student  adhere  to  the 
postulate,  as  universally  valid,  that  water 
runs  down  hill;  or  like  the  confusion  that 
would  arise  in  chemistry  if  the  chemist 
assumed  that  the  rate  of  chemical  reaction 
depends  solely  upon  the  kind,  and  never  upon 
the  amount,  of  the  substances  involved;  or 
like  the  contradictions  that  would  confound 
the  theory  of  functions  if  it  laid  down  as  a 
postulate  that  every  continuous  function 
possesses  a  derivative;  or  like  the  contra- 
dictions that  would  stay  the  progress  of 
geometry  did  this  science  assume  that  all 
geometric  constructions  are  feasible  with 
ruler  and  compasses ;  or,  in  general,  like  the 
entanglements  that  must  always  ensue  when- 
ever, in  any  field  of  thought,  we  consciously 


THE  OLD  THEOLOGY  85 

or  unconsciously  employ  one  or  more  postu- 
lates that,  though  valid  elsewhere,  in  a 
more  restricted  field,  are  not  valid  in  the 
field  of  our  actual  operation.  What  is  the 
remedy?  It  obviously  is  to  reject  the  postu- 
lates whence  the  entanglements  arise.  The- 
ology is,  then,  confronted  with  the  task  of 
weeding  her  garden  of  alien  postulates.  The 
task  is  difficult.  Theology  must  ascertain 
what  her  postulates  are — what  assumptions 
she  actually  makes — and  this  is  not  easy  to 
do,  for  assumptions  are  sly — they  do  not, 
as  a  rule,  loudly  proclaim  their  arrival  or 
their  presence.  Moreover,  when  once  they 
are  ascertained,  there  remains  the  difficult 
problem  of  discrimination:  which  of  them 
are  legitimate,  or  domestic,  and  which  are 
illegitimate,  or  foreign? 

Perhaps  the  most  noxious,  certainly  the 
most  flagrant,  of  theology's  foreign  postu- 
lates— one  that  has  engendered  endless  con- 
fusion within  and  brought  from  without  no 
end  of  ridicule — is  the  hoary  assumption 
that,  in  every  subject-matter  or  field  of 
thought,  the  whole  is  greater  than  the  part. 
It  is  not  perhaps  strange  that  this  so-called 
axiom  became  an  article  of  universal  belief 


86          THE  NEW  INFINITE  AND 

in  the  early  stages  of  human  development, 
when  the  interests  of  men  were  of  necessity 
confined  to  the  concrete  things  of  sense,  but 
it  is  strange,  very  strange,  that  the  belief 
persisted  as  universal  throughout  the  his- 
tory of  thought  despite  the  fact  that  the 
subject-matter  of  thought  is  everywhere  and 
continuously  vocal  with  its  denial.  Except 
in  the  case  of  mathematicians  and  some 
philosophers,  the  proposition  is  even  today 
universally  held  to  be  universally  valid.  The 
fact  is,  however,  as  we  have  abundantly  seen, 
that  the  proposition,  instead  of  being  univer- 
sally true,  is  generally  false.  The  discovery 
of  this  fact — the  discovery,  about  fifty  years 
ago,  that,  instead  of  being  an  essential  prin- 
ciple of  reason,  the  proposition  merely  serves 
as  a  principle  of  classification,  as  a  logical 
blade,  we  may  say,  sundering  the  universe 
of  thinkable  things  into  two  components; 
the  discovery  that  one  of  these — the  world 
of  finite  things — is  composed  of  wholes  to 
which  the  proposition  does  indeed  apply 
without  exception,  but  that  the  other  com- 
ponent— the  world  of  infinites — is  composed 
of  wholes  for  which,  without  exception,  the 
proposition  is  false;  the  discovery  that  the 


THE  OLD  THEOLOGY  87 

latter  world,  the  world  of  infinite  wholes,  is 
par  excellence  the  domain  of  reason,  and 
that,  in  respect  of  content,  it  is  immeasur- 
ably richer  than  the  world  of  finite  wholes : 
that  discovery  I  judge  to  be  second  in 
importance,  for  the  future  of  thought,  to  no 
event  in  the  history  of  mankind.  And 
auspicious  for  theology  will  be  the  day  when 
she  really  discovers  that  Discovery,  when  she 
really  learns  that  her  subject-matter  belongs 
strictly  to  the  world  of  infinite  wholes,  and 
accordingly  relinquishes,  as  then  she  will,  the 
ancient  dogma  of  whole  and  part  as  alien  to 
her  field. 

Let  me  give  an  example  to  illustrate  the 
great  emancipation  that  will  ensue.  Not 
long  ago  in  a  western  city  of  the  United 
States  a  great  orator,  speaking  of  the 
dogma  that  the  persons  of  the  Trinity  are 
each  Almighty  and  yet  together  constitute 
but  one  Almighty,  speaking  of  the  doctrine 
that  each  of  the  Persons  is  equal  to  the  One 
composed  by  all  of  them,  evoked  general 
applause  from  a  vast  audience  by  character- 
izing the  venerated  creed  as  "infinitely 
absurd."  Why?  Because  the  speaker  and 
his  hearers  tacitly  assumed  that  as  a  matter 


88          THE  NEW  INFINITE  AND 

of  course  the  whole  must  exceed  the  part. 
And  why  does  not  theology  explain  the 
difficulty?  Why  does  she  content  herself 
with  avowing  that  the  alleged  composition 
of  the  Trinity  is  an  "incomprehensible 
mystery"?  Because  she,  too,  makes  the 
same  assumption.  And  yet  it  is  not  the 
dogma  but  the  orator's  characterization  of 
it  that  is  "infinitely  absurd."  Let  us  see 
clearly  that  this  is  so.  It  is  plain  that  we 
have  here  to  do  with  the  structure  of  infinite 
manifolds.  More  than  fifty  years  ago,  that 
profound  mathematician,  philosopher  and 
theologian,  Bernhardt  Bolzano,  pointed  out 
that  "there  are  points  of  view  from  which 
we  perceive  in  God  an  infinite  multiplicity 
(unendliche  Vielheit),  and  there  are  no 
other  viewpoints  from  which  we  attribute 
infinity  to  him."  "Ich  sage  nun,"  he  adds 
in  explanation,  "wir  nennen  Gott  unendlich, 
weil  wir  ihm  Krafte  von  mehr  als  einer  Art 
Zugestehen  miissen,  die  eine  unendliche 
Grosse  besitzen.  So  miissen  wir  ihm  eine 
Erkenntnisskraft  beilegen,  die  wahre  All- 
wissenschaft  ist,  also  unendliche  Menge  von 
Wahrheiten,  weil  alle  ueberhaupt,  umfasst, 
und  so  weiter."  The  key-word,  as  the  con- 


THE  OLD  THEOLOGY  89 

text  shows,  is  the  term  unendliche  Menge, 
infinite  ensemble  or  multitude  or  manifold. 
Now  consider,  for  example,  the  following 
infinite  manifolds :  the  totality  E  of  even 
integers,  the  totality  O  of  odd  ones,  the 
totality  F  of  fractions  having  integers  for 
their  terms.  Denote  by  M  the  totality  of 
rational  numbers.  M,  you  see,  is  composed 
of  the  elements  of  E  and  0  and  F.  M  con- 
tains each  of  these  elements  once  and  only 
once  and  contains  nothing  else.  Any  child 
knows  that  there  is  an  even  integer  for  every 
odd  one,  an  odd  integer  for  every  even  one, 
and  so,  it  is  plain,  E  and  O  are  equally  rich 
in  constituents.  Recall,  as  we  proved  some 
pages  back,  that  the  same  is  true  of  E,  which 
we  there  denoted  by  (P),  and  our  old  friend, 
the  ensemble  (W).  Hence  it  is  true  of  O 
and  (  W) .  It  is  true  also  of  F  and  (  W)  for 
we  saw  that  we  could  match  the  integers  with 
the  rational  fractions.  Hence  it  is  true  of 
E  and  O  and  F  and  (  W)  :  these  are  equally 
rich  in  elements.  But  did  we  not  show  the 
like  for  M  and  ( W)  ?  We  did,  whence  it 
follows  that,  in  wealth  of  constituents,  E  and 
O  and  F  and  M  are  exactly  on  a  par:  they 
belong  to  the  same  type  of  infinitude.  It 


90          THE  NEW  INFINITE  AND 

happens  that  it  is  the  denumerable  type  but 
that  fact  is  not  important.  What  is  impor- 
tant is  now  obvious :  it  is  that  we  have  here 
three  infinite  manifolds,  E,  O,  F,  no  two  of 
which  have  so  much  as  a  single  element  in 
common,  and  yet  the  three  together  consti- 
tute one  manifold  M  exactly  equal  in  wealth 
of  elements  to  each  of  its  infinite  components. 
Have  we  proved  that  there  is  a  Trinity 
composed  of  three  components  related  to  one 
another  and  to  the  Trinity  as  the  dogma 
asserts?  No.  We  hayeproved  that  the 
conception  of  such  a  Trinity,  instead  of, 
being  rendered  absurd  by  a  so-called  axiom 
having  no  application  to  infinite  manifolds, 
is  rigorously  thinkable,  perfectly  possible 
and  rational,  and  that  our  brilliant  orator 
was  indeed  in  this  instance  an  ass.  Far  from 
being  absurd,  the  conception  would  be  rigor- 


ously thinkable — as  mathematicians  know 
and  as  the  reader  of  these  pages  ought  now 
to  see — even  if  the  One  it  contemplates  were 
asserted  to  be,  instead  of  a  trinity  of  per- 
sons, a  multiplicity  of  order  four  or  a  mil- 
lion. Nay,  an  infinite  I  of  even  the  lowest 
type  always  contains,  not  merely  two  or  three 
or  a  million  components  each  equal  to  it  in 


THE  OLD  THEOLOGY  91 

plenitude  of  elements,  but  an  infinity  of  such 
components.  The  like  is  equally  true  of  the 
infinites  of  whatever  type  in  the  endless  scale 
of  types.  Must  we  suppose  the  truth  to  fail 
in  the  case  of  theology's  Infinite,  the  great 
ideal  towards  which  the  others  mount  for- 
ever, ever  rising  from  the  level  of  one 
sublimity  to  another  yet  more  sublime?  Is 
the  nature  of  an  ideal  inferior  to  that  of  the 
ideas  it  hovers  above?  Is  perfection  inferior 
to  approximation? 

For  another  example  of  the  great  emanci- 
pation that  will  come  to  theology  the  moment 
she  casts  off  the  yoke  of  the  'whole-part 
axiom'  that  has  hitherto  hampered  the 
proper  movement  of  her  thought,  witness 
the  possibility  of  handling  anew  and  radi- 
cally the  question  of  Omniscience  in  relation 
to  that  of  Freedom.  I  purpose  to  treat  here 
briefly  a  single  phase  of  the  matter,  a  cen- 
tral difficulty  of  it,  familiar  to  every  one. 
If,  so  the  dialectic  runs,  God  is  omniscient, 
he  knows  what  I  shall  do,  he  knows  my 
future,  he  knows,  before  I  make  decisions, 
what  they  will  be,  and  if  he  knows  that,  then 
to  trust  the  feeling  that  I  am  free  to  choose 
is  "to  cheat  the  eye  with  blear  illusion."  On 


92          THE  NEW  INFINITE  AND 

the  other  hand,  if  God  does  not  know  all 
future  events,  he  is  not  omniscient  and  the 
supreme  dignity  ascribed  to  him  is  thereby 
impaired.  The  argument  is  specious  but, 
as  we  are  going  to  see,  it  is  false.  The 
problem  is  to  reconcile,  not  Freedom  and 
Omniscience,  but  Freedom  and  the  Dignity 
of  omniscience.  Let  it  be  granted  that,  if 
you  are  free,  God  is  not  omniscient.  It  does 
not  follow  that  he  is  less  in  respect  of 
dignity  than  if  he  were  omniscient.  Sup- 
pose two  Beings :  one  of  them  capable  at  once 
of  knowing  all  and  of  not  knowing  all;  the 
other  one  capable  of  knowing  all  but  inca- 
pable of  not  knowing  all.  Are  they  coequal 
in  respect  of  dignity  ?  No,  you  will  probably 
say,  the  latter  one  is,  in  respect  of  dignity, 
distinctly  inferior  to  the  former.  If  that  be 
your  answer,  I  shall  agree.  I  do  not,  how- 
ever, intend  to  depend  here  upon  such 
intuitive  estimates  of  worth.  I  purpose  to 
prove  that  a  Being  of  infinite  knowledge  may 
have  all  the  Dignity  of  Omniscience  without 
being  omniscient.  To  do  so,  we  must  again 
have  recourse  to  the  nature  of  infinite  mani- 
folds. Instead,  however,  of  employing,  as 
I  might,  any  of  those  hitherto  presented,  I 


THE  OLD  THEOLOGY  93 

shall  ask  you  to  consider  a  more  shining  one, 
one  that  appeals  to  the  imagination  like  the 
open  sky. 

Let  II  be  an  entire  plane;  it  bisects  the 
universe  of  Space.  I  must  ask  the  reader 
to  assume — for  it  is  true  and  might  easily 
be  shown  did  space  allow — that  a  one-to-one 
correspondence,  of  the  kind  with  which  he  is 
now  familiar,  can  be  established  between  the 
totality  T  of  points  in  space,  those  of  77 
included,  and  the  totality  S  of  points  on 
either  side  of  77.  Note  carefully  that,  as 
77  is  any  plane,  the  correspondence  will  be 
equally  possible,  if  77  be  moved  parallel  to 
itself  any  finite  distance.  Now  suppose  each 
point  of  the  infinite  totality  T  to  represent 
an  element  e  of  knowable  reality,  and  denote 
by  d  the  element  of  spiritual  dignity  that 
attaches  to  knowledge  of  e.  At  once  we  see 
that  a  knowledge  K*  extending  to  all  and  only 
the  elements  e  of  the  part-totality  Se  of 
knowable  reality  represented  by  the  points 
of  S  is  precisely  as  rich  in  elements  d  of 
scientific  or  spiritual  dignity  as  is  a  knowl- 
edge Kt  extending  to  all  the  elements  e  of  the 
whole-totality  Te  of  knowable  reality  repre- 
sented by  the  points  of  T.  Now  suppose  that 


94          THE  NEW  INFINITE  AND 

Te  is  the  whole  of  knowable  reality,  then  Kt 
is  omniscience.  We  behold  the  astounding 
fact  that  omniscience  does  not  by  even  the 
smallest  mite  surpass  in  dignity  the  partial 
knowledge  K8.  But  how,  one  may  ask,  does 
this  fact  advance  the  solution  of  our  prob- 
lem? How  does  it  enable  us  to  maintain  the 
doctrine  of  Freedom  and  still  attribute  to 
God  a  dignity  of  knowledge  equal  to  the 
Dignity  of  omniscience?  For,  our  inter- 
locutor will  say,  knowledge  is  related  to 
Time,  it  is  of  things  that  have  been  or  are 
or  will  be;  omniscience  must  cover  them  all, 
it  must  extend  at  once  through  Past,  Present 
and  Future;  whilst  Freedom  means  that  you 
and  I  are  capable  of  determining  what  the 
Past  is  to  be  by  choosing  in  the  Present,  for 
actualization,  from  among  the  possibilities 
that  constantly  descend  upon  us  out  of  Time- 
to-come  like  in-rolling  waves  of  an  infinite 
sea.  But,  our  critic  will  urge,  such  capa- 
bility does  not  exist  if  omniscience  cover  the 
Future  and  if,  accordingly,  the  destiny  of 
possibilities  is  determined  before  they  arrive. 
And  what,  he  will  say,  is  to  be  said  of  the 
Dignity  of  knowledge  that,  though  covering 
the  Past,  does  not  extend  to  all  events  that 


THE  OLD  THEOLOGY  95 

are  yet  to  be?  In  answer  let  me  ask  the 
reader  to  change  a  little  the  imagery  em- 
ployed in  our  previous  argument:  let  us 
suppose  that  II  is,  not  as  before  an  ordinary 
plane  bisecting  Space,  but  what  we  may  call 
a  moving  Time-plane — the  Present — bound- 
ing off  the  Future  from  the  Past.  Behind  II 
is  an  eternity  of  time  that  has  been;  before 
it,  an  eternity  of  time  that  will  be.  The  two 
eternities,  regarded  as  manifolds  of  the 
things  they  contain,  are  infinitudes  of  the 
same  type,  and — what  is  important  to  note — 
they  are,  as  infinites,  each  of  the  same  type 
as  the  one  Eternity  that  together  they  con- 
stitute. In  respect,  then,  of  Dignity  of 
knowledge,  complete  knowledge  of  the  eternal 
Past  is  not  inferior  to  knowledge  extending 
both  backward  and  forward,  covering  the 
composite  Eternity  of  both  Future  and  Past. 
It  is  important  to  observe  that  the  proposi- 
tion continues  to  be  true  as  the  Time-plane 
77 — advancing  forefront  of  Universal  His- 
tory— with  infinite  range  and  sweep  of  wing 
moves  continuously  forward;  for,  though  the 
Past,  as  we  say,  thus  grows  continuously 
longer  and  longer,  and  the  Future  shorter 
and  shorter,  yet  the  two  eternities  keep 


96          THE  NEW  INFINITE  AND 

forever  their  common  membership  in  the  type 
of  infinity  to  which  they  belong.  And  so  it 
appears  that  Freedom  is  entirely  compatible 
with  the  Dignity  of  omniscience,  though  it 
is  not  compatible  with  Omniscience  itself. 
I  fancy  that  many  a  spiritual-minded  de- 
fender of  the  doctrine  of  Freedom  would 
find  it  no  great  hardship  to  give  up  that  of 
Omniscience,  seeing  that  the  sacrifice  does 
not  involve  denying  to  God  the  Dignity  of 
omniscience.  Such  a  defender  could  say: 
'I  maintain  that,  to  the  Supreme  Intelligence, 
the  Past  alone  is  completely  known;  I  main- 
tain that  the  Future  is  not  completely 
known;  I  maintain  that,  as  the  Present 
moves  on  continuously  forward  into  the 
realm  of  potentialities,  the  eligible  gets 
sifted,  becoming  in  part  the  chosen,  that  part 
of  the  possible  and  unknown  becomes  the 
actual  and  known;  I  maintain  that  mean- 
while the  infinite  Dignity  attaching  to 
knowledge  of  the  growing  Past  remains 
forever  invariant,  equal  absolutely  to  the 
dignity  of  omniscience  itself;  and  that  Free- 
dom remains.'  Many  will  be  glad  to  know 
that  such  a  dogma,  whether  true  or  not,  is 
at  all  events,  thanks  to  the  nature  of  infinite 


THE  OLD  THEOLOGY  97 

manifolds,  free  from  internal  contradiction 
and  may,  therefore,  be  held  without  surren- 
dering reason.  Unless  I  am  much  mistaken, 
the  distinction,  herewith  mathematically 
drawn,  between  the  Dignity  of  omniscience 
and  Omniscience  itself,  whereby  we  may 
affirm  the  doctrine  of  Freedom  without  im- 
puting to  God's  knowledge  a  Dignity  less 
than  that  of  knowing  all,  is  fundamental.  I 
leave  it  to  the  reader  to  see,  in  the  light  of 
his  own  reflection,  that  a  similar  distinction 
is  available,  if  required,  in  dealing  with  other 
attributes — Omnipotence,  for  example,  or 
Omnipresence — commonly  ascribed  to  Deity. 
I  purpose  to  deal  here  with  Omnipresence 
but  from  another  point  of  view. 

Our  task  is  to  vindicate  the  logical  possi- 
bility of  Omnipresence — not  by  such  inade- 
quate analogies  as  immortal  Bruno,  for 
example,  ingeniously  employed  in  comparing 
it  to  a  voice  audible  at  every  point  of  a 
room — but  by  considerations  bringing  it 
strictly  within  the  category  of  doctrines 
rigorously  thinkable.  Consider  a  sphere. 
Let  it  be  so  small  that,  even  if  it  were  a 
brilliantly  colored  globe,  the  most  powerful 
microscope  could  not  reveal  its  presence. 


98          THE  NEW  INFINITE  AND 

It  is  to  be  carefully  noted  that  the  following 
statements  regarding  it  are  absolutely  inde- 
pendent of  its  size,  and  remain  true  if  it  be 
supposed  shrunken  to  any  degree  of  parvi- 
tude,  however  small,  so  long  as  it  has  not 
vanished  utterly.  Denote  by  *  the  totality 
of  points  within  the  tiny  sphere,  and  by  S 
the  ensemble  of  all  the  other  points  of  the 
whole  of  Space.  In  the  course  of  recent 
years  and  by  means  within  the  grasp  of  the 
average  student  a  little  disciplined  in  the 
ways  of  rigorous  thought,  it  has  been  demon- 
strated that  there  are  precisely  as  many 
points  in  s,  as  in  S,  and  that  the  former  are 
joined  to  the  latter  in  one-to-one  fashion 
by  relational  rays  of  correspondence.  As 
such  correlation  subsists  in  countless  modes, 
suppose  one  of  them  chosen.  This  done,  to 
any  point  of  S,  say  the  center  of  the  sun, 
corresponds  a  definite  point  of  *;  to  any 
other  point  of  S,  say  the  center  of  the  moon 
or  the  mass-center  of  the  Milky  Way,  corre- 
sponds another  definite  point  of  s;  and  so  on 
and  on  throughout  the  range  of  both  totali- 
ties :  no  element  of  either  manifold  but  it  has 
a  match  or  mate  in  the  other  and  no  two 
of  either  manifold  have  a  common  mate. 


THE  OLD  THEOLOGY  99 

Let  no  one  fail  to  see  clearly  that  in  that 
tiny  sphere,  too  small,  mind  you,  for  even 
microscopic  vision,  small  indeed  at  will, 
there  nevertheless  exist  point  configurations 
matching  perfectly  in  detail  and  every  re- 
spect of  inner  constitution  each  and  all  of 
the  infinitely  infinite  hosts  of  point  configu- 
rations, minute  and  vast,  simple  and  com- 
plex, here,  there,  and  yonder,  everywhere 
throughout  the  height  and  depth  and  length 
and  breadth  of  Space.  We  have  now  only 
to  reflect  that  the  same  scheme  of  repre- 
sentation obtains  universally,  being  valid  at 
once  for  all  infinitesimal  spheres,  and  the 
truth  dawns  that  the  Whole  really  is  incar- 
nate in  every  Part — the  Emersonian  apho- 
rism that  "the  universe  contrives  to  integrate 
itself  in  every  smallest  particle"  being  thus 
completely  justified  on  scientific  ground. 
But  this  is  yet  not  all.  The  universe  is 
dynamic,  charged  throughout  with  innumer- 
able modes  of  motion.  Each  point,  however, 
of  any  moving  thing — an  ion  of  gas,  a 
vibrating  fiber  of  brain — is  represented  by 
a  corresponding  point  in  s,  and  so  within 
the  tiny  sphere — indeed  in  every  room  how- 
ever small — the  whole  dynamics  of  the  uni- 


100        THE  NEW  INFINITE  AND 

verse  is  depicted  completely  and  coenacted 
by  motion  of  points  and  transformation  of 
point  configurations.  There  in  miniature 
proceed  at  once  the  countless  play  and  inter- 
play of  every  kind  of  motion,  small  and 
large,  simple  and  complex,  the  quivering 
dance  of  the  molecule,  the  wave  and  swing 
of  universal  aether. 

"  Wie  Alles  sich  zum  Ganzen  webt ! 
Eins  in  dem  andern  wirkt  und  lebt! 
Wie  Himmelskrafte  auf  und  nieder  steigen 
Und  sich  die  goldnen  Eimer  reichen! 
Mit  segenduftenden  Schwingen 
Vom  Himmel  durch  die  Erde  dringen, 
Harmonisch  all'  das  All  durchdringen !" 

The  immense  labor  to  be  performed  by 
theology  in  eradicating  from  the  proper 
domain  of  her  study  the  whole-part  dogma 
with  its  ubiquitous  progeny  of  confusion ;  and 
the  light,  the  freedom,  and  the  power  that 
will  more  and  more  accrue  to  her  as  the 
work  proceeds:  these  are  not  the  end  but 
are  only  the  beginning  of  her  emancipation. 
For  the  whole-part  "axiom"  is  not  the  sole 
postulate  of  the  imported  kind  that  troubles 
her  thought.  Once  she  seriously  enters  upon 
the  search,  she  will  find  that  there  are  others. 
I  have  already  repeatedly  pointed  out  that 


THE  OLD  THEOLOGY  101 

the  subject-matter  of  her  thought — the 
realm  of  transfinite  reality — presents  infini- 
tudes in  a  hierarchy  without  a  summit.  As 
she  passes  upward  in  her  study  from  level 
to  level,  she  will  find  that  a  postulate  avail- 
able at  a  given  elevation  may  have  to  be 
relinquished  on  passing  to  a  higher  rank. 
For  example,  nothing  can  seem  more  natural 
or  axiomatic  than  to  suppose  that,  if  we 
have  any  manifold  of  elements,  these  are 
capable  of  being  arranged  in  a  row,  like 
marbles,  so  that  after  each  there  is  a  next — 
none,  that  is,  between.  Nevertheless,  as 
mathematicians  have  recently  ascertained, 
that  seemingly  universal  possibility  is  re- 
stricted very  narrowly.  The  possibility — 
let  us  call  it  the  postulate  of  Nextness — 
does  indeed  hold  for  all  infinite  manifolds  of 
the  Denumerable  type  but  it  fails  utterly 
for  every  manifold  of  the  Continuum  type 
or  of  any  higher  type.  The  employment  of 
foreign  postulates  is  equally  disastrous 
whether  the  importation  be,  as  in  case  of  the 
whole-part  postulate,  from  the  realm  of  the 
finite,  where  it  is  valid,  to  the  realm  of  the 
infinite,  where  it  is  not,  or,  as  in  case  of  the 
nextness  postulate,  from  a  type  of  infinitude, 


102        THE  NEW  INFINITE  AND 

in  which  it  applies,  to  a  higher  type,  in  which 
it  does  not.  It  is  now,  I  believe,  sufficiently 
evident  that  eternal  vigilance  against  the 
admission  of  alien  assumptions  is  part  of 
the  price  theology  must  pay  for  freedom, 
for  freedom,  that  is,  from  the  fatal  presence 
of  internal  confusion. 

Before  undertaking  to  deal  with  the  other 
variety  of  contradictions — the  kind,  I  mean, 
that  arise  properly,  because  they  arise  from 
domestic  or  native  postulates — ,  I  desire  to 
allude  briefly  to  another  mathematical  idea, 
one  that  is  destined,  I  believe,  as  the  eye 
becomes  more  and  more  adjusted  to  its  light, 
to  be  of  great  service  in  theology,  especially 
enlarging  her  conception  of  the  Conceivable, 
and  serving  to  bring  not  only  the  attribute 
of  Omnipresence,  with  which  we  are  here 
further  concerned,  but  kindred  attributes  as 
well,  strictly  within  the  category  of  intelli- 
gible ideals.  I  refer  to  the  radiant  concept 
of  Hyperspace.  Only  a  generation  ago  this 
concept  was  regarded  even  by  mathemati- 
cians— most  adventurous  of  men — as  vision- 
ary and  vain.  Meanwhile  it  has  advanced 
so  rapidly  to  commanding  position  that 
today  its  instrumental  value  is — strange  to 


THE  OLD  THEOLOGY  103 

say — recognized  even  in  "natural"  science, 
by  Van't  Hoff,  for  example,  in  chemistry, 
and  by  leading  physicists  in  the  kinetic 
theory  of  gases.  The  statement  made  by 
Poincare  seven  years  ago  before  the  Inter- 
national Congress  of  Mathematicians  at 
Rome  is  well  within  conservative  limits : 
"Nous  sommes  aujourd'hui  tellement  famil- 
iarises avec  cetti  notion  que  nous  pouvons 
en  parler,  meme  dans  un  cours  d'universite, 
sans  provoquer  trop  d'etonnement."  The 
fact  is  that  the  doctrine  of  hyperspaces 
already  exists  in  a  copious  and  rapidly 
growing  literature,  flourishes  in  every  scien- 
tific language  of  the  world,  and  in  its  essen- 
tial principles  has  become  for  mathematicians 
as  orthodox  as  the  multiplication  table. 
Indeed,  as  Professor  Klein  has  shown,  the 
modern  physical  theory  of  Relativity  is,  in 
point  of  structure  and  form,  a  species  of 
four-dimensional  geometry.  My  aim  here  is 
to  indicate  how  the  hyperspace  concept 
enables  us  to  show  the  conceivability  of  an 
infinite  Being  being  everywhere  present  in 
an  infinite  region  without  being  contained  in 
it.  Anyone  who  will  devote  a  little  time  to 
reflecting  upon  the  infinite  wealth  of  points 


104        THE  NEW  INFINITE  AND 

in,  say,  a  straight  line  L  and  upon  the  infinite 
wealth  of  detectible  combinations  and  inter- 
relations subsisting  among  them,  will  dis- 
cover to  his  astonishment  that  a  linear  being 
or  intelligence  X  inhabiting  L  and  in  its  expe- 
rience strictly  confined  thereto  would  have, 
in  its  own  habitation,  all  the  material 
necessary  for  constructing  mathematical  doc- 
trines matching  completely,  in  diversity 
and  in  complexity,  all  branches  of  geometry 
and  analysis  constructible  by  man,  despite 
the  immensely  superior  resources  the  latter 
seems  to  have  in  inhabiting  the  three-way 
spread  of  Space.  Marvelous  as  it  seems,  the 
parity  exists.  Such  a  being  X,  dwelling  in 
the  midst  of  such  magnificence  of  subject- 
matter,  order,  and  law,  naturally  might 
attempt  to  construct  a  rational  theology. 
If  so,  it  would  encounter,  among  other  diffi- 
culties, that  of  understanding  how  the 
supreme  being  it  hypothetized  could  be  at 
one  and  the  same  time  present  everywhere  in 
the  line-world  L.  Note  that,  by  hypothesis, 
X  could  have  no  sense-perception  or  geometric 
intuition  or  image  of  the  fact  that  the  infinite 
region  or  line-world  L,  in  which  it  lives, 
moves  and  has  its  being,  is,  as  we  humans 


THE  OLD  THEOLOGY  105 

happen  to  know,  itself  contained  or  immersed 
in  another  infinite  region  of  higher  order, 
namely,  a  plane  II;  hence  X  could  not  per- 
ceive, though  it  might  feel,  and  it  might  in- 
deed conceive,  the  fact  that  the  infinite,  77,  is 
actually  omnipresent  to  L,  every  part  of  this 
line- wo  rid  being,  as  we  know,  completely 
immersed  in  II;  and  so  A  could  not  perceive, 
yet  after  some  centuries  of  theologizing  it 
might  cowceive,  how  the  same  attribute — 
omnipresence  in  the  line-world  L — could 
belong  to  a  being  whose  reality,  whatever  its 
nature  in  other  respects,  was  at  least  co- 
extensive with  the  higher  world  II.  Who 
can  fail  to  see  that  precisely  like  reflections 
would  be  equally  pertinent,  if  we  replaced 
the  line- wo  rid  L  by  the  plane- world  77  and 
the  latter  by  Space  itself?  We  live  in 
Space — a  three-way  spread — and  encounter 
precisely  the  same  difficulties  encountered  by 
our  linear  friend  A,  and  they  are  surmount- 
able in  the  same  way,  namely,  by  the  concept 
of  Hyperspace.  For  this  world-creating 
idea,  at  once  exquisite  and  vast,  presents  us 
in  the  first  place  with  a  four-way  spread,  a 
four-dimensional  space,  >S4,  completely  im- 
mersing our  ordinary  space,  being  in  contact 


106        THE  NEW  INFINITE  AND 

with  all  its  points  and  present  at  all  of  them, 
just  as  our  ordinary  space  is  omnipresent 
to  all  the  elements  of  the  plane-world  77,  and 
this,  in  turn,  to  all  those  of  L;  next,  similarly 
related  to  S4,  comes  a  yet  higher  world  S5; 
then  follow,  in  order  of  ascending  dimension- 
ality, iS6,  iS7,  .  .  .,  •S'n,  .  .  .  and  so  on  end- 
lessly: affording  thus  conceptual  provision 
for  the  presence  everywhere  in  our  dwelling- 
place  of  a  Being  whose  reality,  if  you  please, 
not  only  pervades  but  infinitely  transcends 
any  assignable  space,  however  high  its  rank 
in  the  summitless  scale  of  hyperspatial 
grandeur.  Is  it  a  small  service  to  show  that 
theology's  supreme  ideals  conform  to  pat- 
terns woven  of  scientific  ideas?  Is  it  a  little 
thing  to  demonstrate  the  reasonableness  of 
reason's  dreams? 

Finally,  I  come  now  to  the  keeping  of  my 
promise  regarding  theological  difficulties  of 
the  domestic  kind.  These  are  not  due  to  the 
lurking  presence  of  alien  postulates,  and  are 
not  to  be  overcome  by  the  process  of  casting 
out.  They  are  due  to  the  peculiarly  vast 
and  complicate  character  of  theology's 
subject-matter,  to  the  great  diversity  of 
aspects  presented  by  it,  and  the  consequent 


THE  OLD  THEOLOGY  107 

necessity  of  examining  or  beholding  them 
from  a  corresponding  variety  of  partial  or 
fragmentary  points  of  view.  Such  native 
difficulties  are  to  be  conquered,  progressively 
of  course,  not  by  elimination,  but  by  the 
method  of  surmounting,  by  the  process  of 
transcending.  What  does  this  method  con- 
sist in?  What  does  such  transcending  mean? 
It  does  not  mean,  as  commonly  supposed,  the 
finding  of  a  point  of  view  from  which  the 
difference  of  two  aspects  of  a  matter  shall, 
as  this  is  seen  from  other  points  of  view, 
seem  to  disappear,  for  that  would  be,  not  to 
clarify,  but  to  obscure,  to  disguise  fact,  to 
hide  truth.  Transcending  does  not  mean 
that.  It  means — and  the  answer  is  very 
important — recognition  of  the  fact  that  two 
differing  aspects  of  a  matter  are  indeed,  not 
one,  but  two,  and  that  the  matter  is,  in  truth, 
such  as  to  present  them  both.  It  thus  means 
submission  of  the  understanding  to  facts, 
not  facts  to  the  understanding,  and,  in  dis- 
course, to  speak  of  a  matter  as  it  is  and  not 
as  we  may  wish  it  to  be.  Doubtless  the  aim 
of  science  is  art  but  the  beauty  it  seeks  does 
not  lie  in  the  way  of  disguisings  or  mutila- 
tions, for  it  is  the  beauty  of  truth. 


108        THE  NEW  INFINITE  AND 

Before  presenting  concrete  illustrations 
may  I  outline  the  matter  briefly  in  abstract? 
Denote  by  B  some  being,  some  complex  and 
multi-phased  entity,  the  subject  or  object  of 
thought.  In  view  of  some  aspect  of  B  we 
construct  a  theory  TI,  which,  as  we  are  not 
aware  of  other  aspects,  we  call  a  theory,  not 
of  a  phase  of  B,  but  of  B  itself.  Some  other 
aspect  of  B,  seen  at  another  time  by  us  or 
at  the  same  time  by  some  one  else,  gives  rise 
to  another  theory  T2,  which,  like  7\  and 
owing  to  the  same  circumstance,  claims  to  be 
a  theory  of  B;  and  so  on,  for  other  phases  of 
B.  Let  us  suppose  that  the  theories  have 
been  soundly  made  after  the  manner  of 
autonomous  doctrines.  7\,  then,  consists 
of  a  definite  basal  system  of  compatible 
postulates  together  with  a  superstructure  of 
rigorously  deduced  implications.  Of  T2,  we 
must  say  the  same.  Each  of  the  theories  is, 
accordingly,  thoroughly  coherent,  absolutely 
void  of  inconsistency  among  its  component 
elements.  They  do  not,  however,  coincide: 
though  having  perhaps  many  propositions 
in  common,  yet  either  T  contains  at  least 
one  proposition  that  contradicts  some  propo- 
sition of  the  other.  Let  us  suppose,  more- 


THE  OLD  THEOLOGY  109 

over,  that  each  theory  is  true  to  the  aspect 
that  gave  it  birth:  that  is,  seen  from  one 
point  of  view,  B  appears  exactly  as  7\ 
describes  it;  from  another,  exactly  as  Tz 
describes  it ;  and  so  on,  of  course,  if  there  be 
other  theories.  What  happens?  This: 
sooner  or  later,  in  one  or  another  of  the  ways 
familiar  to  students,  7\  and  Tz  get  com- 
pared; it  is  noted  that  each  of  them  claims 
to  be  a  true  theory  or  account  of  B;  it  is 
observed  also  that  in  one  or  more  respects 
they  are  mutually  contradictory.  What 
follows?  It  follows  that  the  claim  must  be 
disallowed  in  the  case  of  at  least  one  of  them : 
regarded  as  accounts  of  one  object  or  sub- 
ject, two  discordant  doctrines  may  be  both 
of  them  false  but  they  can  not  both  be  true. 
But  we  have  seen  that  each  of  them  is  true 
to  the  B-aspect  that  gave  it  birth;  yet  they 
contradict  one  another.  What  is  to  be  done? 
Reject  them  both?  No.  The  remedy  is: 
Keep  both  and  transcend  them.  Keep  them 
both,  for  the  contradiction  arises  from  sup- 
posing them  to  be  speaking  of  B  as  a  whole, 
which  they  are  really  not;  it  disappears 
when  we  suppose  them  to  be  speaking 
respectively  of  different  phases  of  B,  which 


110        THE  NEW  INFINITE  AND 

they  really  are.  The  act  or  process  of  sur- 
mounting consists — not  in  constructing  one 
theory  to  cover  at  once  both  the  aspect 
covered  by  TI  and  that  covered  by  T2,  for 
that  is  impossible — the  nearest  possible  ap- 
proach to  it  would  be  to  construct  a  theory 
covering  the  common  part  (if  any)  of  the 
two  aspects,  and  plainly  such  a  theory  would 
consist  of  the  common  part,  or  intersection, 
of  TI  and  T2:  no,  the  surmounting  or  tran- 
scending of  T!  and  T2  consists  in  recognizing 
once  for  all  that  the  object  B  does  in  fact 
present  the  two  aspects  in  question  and 
thereby  validates  at  once  both  of  the  theories 
in  question.  Do  you  ask  what  is  thus  gained? 
I  answer  that  the  proposition  stating  the 
recognition  is  new;  it  is  not  in  Tx  nor  in  T2; 
it  is  not  a  fact  about  either  of  the  phases 
dealt  with  by  7\  and  T2;  we  have  mounted 
higher — the  new  truth  is  a  truth  about  B 
itself. 

Is  the  matter  so  clear  in  the  abstract  as 
not  to  be  impressive?  I  sincerely  hope  that 
it  is,  for  in  that  case  it  will  not  require  many 
concrete  illustrations  and  of  these,  moreover, 
the  simplest  will  suffice.  I  will  begin  with  one 
so  simple  as  to  seem  trivial.  Yet  its  illustra- 


THE  OLD  THEOLOGY  111 

tive  value  is,  I  believe,  very  considerable, 
unless  our  familiarity  with  the  phenomena 
involved,  blinds  us  to  their  worth.  On  my 
table  lies  a  slender  rod.  As  seen  there,  it 
appears  to  be  straight.  I  place  it  at  a  slant 
in  a  vessel  of  water.  As  seen  there,  it  ap- 
pears to  be  bent.  Is  the  rod  straight  or 
bent?  That  is  not  the  question.  If  it  were, 
we  should  have  to  invoke  the  testimony  of  at 
least  another  sense,  which,  however,  for  the 
purpose  of  the  illustration,  I  exclude.  I  am 
admitting  vision  only.  To  vision,  then,  the 
rod  presents  two  contradictory  aspects — 
now  straight,  now  bent.  Are  they,  as 
aspects,  false?  Is  either,  as  an  aspect,  false? 
Neither,  as  an  aspect,  is  false:  as  aspects, 
both  are  true,  both  are  genuine,  both  actual. 
How  surmount  them?  The  answer  is  by 
recognizing  that  the  rod  is  such  a  thing  in 
our  world  that  it  does,  in  truth,  present  to 
vision  both  aspects — and  that  recognition  is 
a  valuable  event  because  it  tells  a  truth  about 
the  rod,  about  our  world,  and  about  our 
vision. 

For  another  lean  but  helpful  illustration, 
consider  the  quadratic  expression,  x2 —  4. 
If  you  write,  x2 — 4  =  0,  then  I  can  affirm 


112        THE  NEW  INFINITE  AND 

that  x  =  2  or  that  a;  —  — 2  but  you  are 
right  in  not  allowing  me  to  say  that  x  =  both 
2  and  — 2  at  once.  Everyone  knows,  how- 
ever, how  to  transcend  the  seeming  necessity 
of  the  alternation:  x  =  2  or  a:  =•  — 2.  We 
do  it,  that  is,  by  saying  that  the  given 
equation  is  a  thing  of  which  2  and  — 2  are, 
at  the  same  time,  roots.  Is  such  surmounting 
merely  a  trick?  On  the  contrary  it  is  a 
legitimate  procedure  of  thought:  the  taking 
of  both  of  two  things  when  either  is  allowed, 
or  taking  all  of  many  when  any  is  allowed: 
it  is  the  familiar  bound  of  the  spirit  from 
alternation  to  conjunction  or  more  often 
from  the  level  of  partial  dissonance  to  the 
bridge  of  an  overarching  harmony. 

A  much  more  impressive  example  of  such 
surmounting  is  found  in  the  manner  in  which 
geometricians  deal  with  the  infinitely  distant 
region  of  space.  There  are,  as  the  reader 
may  know,  various  kinds  of  geometry  of 
space.  In  one  of  these  the  infinite  region 
of  space  presents  one  aspect;  in  a  second, 
a  second  aspect;  in  a  third,  a  third;  and  so 
on  indefinitely.  These  various  aspects  differ 
among  themselves  immensely — they  are  even 
inconsistent  with  one  another  or  mutually 


THE  OLD  THEOLOGY  113 

contradictory  and  exclusive.  Thus,  in  what 
is  called  projective  geometry,  alluded  to 
previously  herein,  the  aspect  presented  by 
the  infinite  region  of  space  is  that  of  a  plane 
all  of  whose  points  are  infinitely  far  away; 
in  what  is  called  inversion  geometry,  which 
I  need  not  here  explain,  the  same  infinite 
region  appears  to  be  a  point  where,  curiously 
enough,  all  lines  of  space  seem  to  meet  and 
pass.  What  do  geometricians  do  in  the 
matter  of  such  conflicts,  at  first  so  shocking? 
Do  they  reject  the  aspects  as  false  because 
they  are  mutually  incompatible?  Far  from 
rejecting  any  of  them,  they  keep  them  all, 
use  them  all,  rejoice  in  them  all,  and — tran- 
scend them  all.  But  how  transcend?  Again 
the  answer  is — and  how  replete  with  sig- 
nificance for  theology! — the  answer  is  that 
geometricians  simply  recognize  that  the 
infinitely  distant  portion  of  space  is,  whether 
one  likes  it  or  not  (and  geometricians  do  like 
it),  in  its  own  nature  just  such  a  thing  as 
to  present  in  fact  all  the  diverse  aspects  in 
question,  and  so  to  validate — at  once,  mind 
you — all  the  geometries  in  question. 

Similar  matter  presses   from  every   side; 
but  enough  has  been  said,  I  trust,  to  indicate 


114        THE  NEW  INFINITE  AND 

the  method  that  mathematics  would  recom- 
mend for  dealing  with  theological  difficulties 
of  the  domestic  or  native  kind.  As  theology 
proceeds  with  her  great  enterprise  of  advanc- 
ing the  science  of  Idealization,  as  in  particu- 
lar she  continues  to  clarify  and  estimate  the 
significance  of  the  supernal  ideals  that  she 
ascribes  as  attributes  to  Deity,  she  is  des- 
tined to  discover  that  those  attributes,  how- 
ever indubitable  or  undeniable  they  may  be 
when  regarded  singly,  yet,  taken  together, 
involve  essential  and  ineradicable  incompati- 
bilities of  thought,  and,  therefore,  must 
finally  defeat  every  possible  effort  to  com- 
bine them  in  one  self-consistent  body  of  doc- 
trine. The  question  is,  What  is  to  be  done  in 
that  event?  Answering  out  of  the  fullness 
of  her  own  experience  in  such  cases,  Mathesis 
will  venture  to  offer  her  sister  the  following 
counsel.  "My  years  and  station,"  she  will 
say  to  Theology,  "and  the  character  of  my 
occupation  entitle  me  to  believe  that  I  am 
not  without  some  insight  into  the  nature  of 
your  gravest  difficulty  and  not  without  some 
knowledge  of  the  means  available  for  over- 
coming it.  Usus,  magister  egregius,  hoc  me 
docuit.  I,  too,  in  the  course  of  my  long 


THE  OLD  THEOLOGY  115 

career  have  expended,  I  do  not  say  have 
wasted,  much  time  and  energy  in  attempting 
to  combine  the  non-combinable,  in  attempt- 
ing, that  is,  to  erect  a  solid  and  unitary 
doctrine  respecting  some  object  of  my 
thought  upon  a  basis  of  postulates  that 
were  indeed  individually  sound  and  eligible, 
but  that,  taken  collectively  as  a  system, 
were  subsequently  found  to  involve  logical 
incompatibility  and  so  not  to  allow  any 
superstructure  not  doomed  to  quick  decay 
by  the  presence  within  it  of  fatal  contra- 
dictions. Fortunately,  I  have  not  besought 
or  trusted  any  hyperlogical  providence  to 
preserve  such  architecture  against  external 
criticism  or  the  destructive  agency  of  its 
own  defects,  but  have  had  the  grace  to  tear 
it  down  myself  and  prepare  to  build  anew. 
My  practice  has  been  to  examine  again  and 
patiently  to  reexamine  the  basal  postulates, 
to  form  from  them  by  trial  and  experiment 
as  many  subgroups  as  possible,  subject  to 
the  condition  that  each  of  these  be  entirely 
free  of  interior  inconsistence,  and  then,  upon 
the  subgroups  as  distinct  though  related 
foundations,  to  construct  as  many  distinct 
but  kindred  doctrines,  each  of  strength  to 


116        THE  NEW  INFINITE  AND 

mock  at  time  and  endure  for  aye.  And  my 
practice,  as  you  and  all  the  world  may  know, 
has  been  justified  of  its  fruits.  Examples 
abound  in  every  division  of  my  common- 
wealth, and  some  have  come  to  fame.  To 
cite  but  three  of  these — behold  the  noble 
structures  of  Euclid,  of  Bolyai  and  Lobat- 
schevski,  and  of  Riemann.  There  stand  the 
great  geometries,  each  upon  its  own  founda- 
tion of  compatible  postulates,  and  there, 
flawless  within,  unassailable  from  without, 
they  will  stand  for  ever,  eternal  witnesses 
of  the  fact  that,  contrary  to  many  a  ven- 
erated but  shallow  creed,  one  object  of 
thought  may,  by  virtue  of  its  kind  and  not 
of  limitations  of  the  human  mind,  transcend 
the  bounds  of  any  one  constructible  theory, 
and  in  its  own  ultimate  nature  allow  and 
validate  at  once,  without  annulling  their 
differences,  a  class  of  dissonant  doctrines. 
Thus  you  perceive,  for  example,  that  my 
Geometry  is  one,  though  my  geometries  are 
many — just  as  Music  is  one,  though  its 
forms  be  as  varied  as  the  moods  of  the  sea. 
And  I,  Mathesis,  am  one,  as  Poetry  is  one, 
though  my  theories,  my  doctrines,  are  legion ; 
for  these  but  differ  among  themselves,  as  the 


THE  OLD  THEOLOGY  117 

myriad  forms  of  Art :  each  is  assertable,  each 
being  valid,  of  one  great  Form  common  to 
them  all.  My  meaning,  I  trust,  is  clear. 
Conquest  of  your  gravest  difficulty  demands 
division.  By  the  method  of  trial  and  experi- 
ment, the  fundamental  attributes  that  you 
hypothetise  of  Deity  must  be  assorted  into 
sets  each  composed  of  harmonious  elements. 
Implicit  in  each  such  group  is  a  coherent  and 
sacred  doctrine.  As  these  doctrines  unfold, 
your  conception  of  yourself  will  change :  you, 
Theology,  will  indeed  be  one;  but  many 
your  theologies.  And  thenceforth  the  Object 
of  all  your  thought  will  appear  to  you  and 
will  be  shown  by  you  to  the  world,  not  in  the 
light  of  a  solitary  sun,  but  in  that  of  a 
constellation." 


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